\(x^{2}+14 x+49\)
\(=(x)^{2}+2 \cdot x \cdot 7+(7)^{2}\)
\(=(x+7)^{2}\)
\(=(x+7)(x+7)\)

\(4 m^{2}-36 m+81\)
\(=(2 m)^{2}-2\cdot 2 m \cdot 9+(9)^{2}\)
\(=(2 m-9)^{2}\)
\(=(2 m-9)(2 m-9)\)

\(25 x^{2}+30 x+9\)
\(=(5 x)^{2}+2\cdot 5 x \cdot 3+(3)^{2}\)
\(=(5 x+3)^{2}\)
\(=(5 x+3)(5 x+3)\)

\(121 b^{2}-88 b+16\)
\(=(11 b)^{2}-2 \cdot 11 b \cdot 4+(4)^{2}\)
\(=(11 b-4)^{2}\)
\(=(11b-4)(11 b-4)\)

\(\left(x^{2} y\right)^{2}-4 x^{2} y^{2}\)
\(=x^{4} y^{2}-4 x^{2} y^{2}\)
\(=x^{2} y^{2}\left(x^{2}-4\right)\)
\(=x \cdot x \cdot y \cdot y \cdot\left\{(x)^{2}-(2)^{2}\right\}\)
\(=x \cdot x \cdot y \cdot y \cdot(x+2)(x-2)\)

\(a^{4}+4 a^{2} b^{2}+4 b^{4}\)
\(=\left(a^{2}\right)^{2}+2 \cdot a^{2} \cdot 2 b^{2}+\left(2 b^{2}\right)^{2}\)
\(=\left(a^{2}+2 b^{2}\right)^{2}\)
\(=\left(a^{2}+2 b^{2}\right)\left(a^{2}+2 b^{2}\right)\)

\(4 x^{2}-16\)
\(=4\left(x^{2}-4\right)\)
\(=2\cdot 2 \cdot\left\{(x)^{2}-(2)^{2}\right\}\)
\(=2\cdot 2(x+2)(x-2)\)

\(121-36 x^{2}\)
\(=(11)^{2}-(6 x)^{2}\)
\(=(11+6 x)(11-6 x)\)

\(x^{2} y^{2}-p^{2} q^{2}\)
\(=(x y)^{2}-(p q)^{2}\)
\(=(x y+p q)(x y-p q)\)

\(80 m^{2}-125\)
\(=5\left(16 \mathrm{m}^{2}-25\right)\)
\(=5\left\{(4 \mathrm{m})^{2}-(5)^{2}\right\}\)
\(=5(4 \mathrm{m}+5)(4 \mathrm{m}-5)\)

\(a x^{2}-a y^{2}\)
\(=a\left(x^{2}-y^{2}\right)=a\left\{(x)^{2}-(y)^{2}\right\}\)
\(=a(x+y)(x-y)\)

\(1-(m+n)^{2}\)
\(=(1)^{2}-(m+n)^{2}\)
\(=(1+m+n)(1-m-n)\)

\((2 a-b-c)^{2}-(a-2 b-c)^{2}\)
\(=(2 a-b-c+a-2 b-c)(2 a-b-c-a+2 b+c)\)
\(=(3 a-3 b-2 c)(a+b)\)

\(x^{2}-2 x y-3 y^{2}\)
\(=(x)^{2}-2 \cdot x \cdot y+(y)^{2}-(y)^{2}-3 y^{2}\)
\(=(x-y)^{2}-4 y^{2}\)
\(=(x-y)^{2}-(2 y)^{2}\)
\(=(x-y+2 y)(x-y-2 y)\)
\(=(x+y)(x-3 y)\)

\(x^{2}+9 y^{2}+6 x y-z^{2}\)
\(=\left\{(x)^{2}+2 \cdot x \cdot 3 y+(3 y)^{2}\right\}-z^{2}\)
\(=(x+3 y)^{2}-(z)^{2}\)
\(=(x+3 y+z)(x+3 y-z)\)

\(a^{2}-b^{2}+2 b c-c^{2}\)
\(=a^{2}-\left(b^{2}-2 b c+c^{2}\right)\)
\(=(a)^{2}-(b-c)^{2}\)
\(=(a+b-c)(a-b+c)\)

\(a^{2}(b-c)^{2}-b^{2}(c-a)^{2}\)
\(=\{a(b-c)\}^{2}-\{b(c-a)\}^{2}\)
\(=(a b-a c)^{2}-(b c-a b)^{2}\)
\(=(a b-a c+b c-a b)(a b-a c-b c+a b)\)
\(=(b c-a c)(2 a b-a c-b c)\)
\(=c(b-a)(2 a b-a c-b c)\)

\(x^{2}-y^{2}-6 y z-9 z^{2}\)
\(=x^{2}-\left(y^{2}+6 y z+9 z^{2}\right)\)
\(=(x)^{2}-\left\{(y)^{2}+2 \cdot y \cdot 3 z+(3 z)^{2}\right\}\)
\(=(x)^{2}-\{y+3 z)^{2}\)
\(=(x+y+3 z)(x-y-3 z)\)

\(x^{2}-y^{2}+4 x-4 y\)
\(=(x+y)(x-y)+4(x-y)\)
\(=(x-y)(x+y+4)\)

\(a^{2}-b^{2}+c^{2}-d^{2}-2(a c-b d)\)
\(=a^{2}-b^{2}+c^{2}-d^{2}-2 a c+2 b d\)
\(=a^{2}-2 a c+c^{2}-b^{2}+2 b d-d^{2}\)
\(=(a)^{2}-2 \cdot a \cdot c +(c)^{2}-\left(b^{2}-2 b d+d^{2}\right)\)
\(=(a-c)^{2}-\left\{(b)^{2}-2 \cdot b \cdot d+(d)^{2}\right\}\)
\(=(a-c)^{2}-(b-d)^{2}\)
\(=(a-c+b-d) \cdot(a-c-b+d)\)
\(=(a+b-c-d)(a-b-c+d)\)

\(2 a b-a^{2}-b^{2}+c^{2}\)
\(=c^{2}-a^{2}+2 a b-b^{2}\)
\(=(c)^{2}-\left(a^{2}-2 a b+b^{2}\right)\)
\(=(c)^{2}-\left\{(a)^{2}-2 \cdot a \cdot b+(b)^{2}\right\}\)
\(=(c)^{2}-(a-b)^{2}\)
\(=(c+a-b)(c-a+b)\)
\(=(a-b+c)(b-a+c)\)

\(36 x^{2}-16 a^{2}-24 a b-9 b^{2}\)
\(=36 x^{2}-\left(16 a^{2}+24 a b+9 b^{2}\right)\)
\(=(6 x)^{2}-\left\{(4 a)^{2}+2\cdot 4 a \cdot 3 b+(3 b)^{2}\right\}\)
\(=(6 x)^{2}-(4 a+3 b)^{2}\)
\(=(6 x+4 a+3 b)(6 x-4 a-3 b)\)

\(a^{2}-1+2 b-b^{2}\)
\(=(a)^{2}-\left\{(1)^{2}-2 b+(b)^{2}\right\}\)
\(=(a)^{2}-(1-b)^{2}\)
\(=(a+1-b)(a-1+b)\)
\(=(a-b+1)(a+b-1)\)

\(a^{2}-2 a-b^{2}+2 b\)
\(=a^{2}-2 a+1-b^{2}+2 b-1\)
\(=\left(a^{2}-2 a+1\right)-\left(b^{2}-2 b+1\right)\)
\(=(a-1)^{2}-(b-1)^{2}\)
\(=(a-1+b-1)(a-1-b+1)\)
\(=(a+b-2)(a-b)\)

\(\left(a^{2}-b^{2}\right)\left(c^{2}-d^{2}\right)-4 a b c d\)
\(=a^{2} c^{2}-b^{2} c^{2}-a^{2} d^{2}+b^{2} d^{2}-4 a b c d\)
\(=a^{2} c^{2}-2 a b c d+b^{2} d^{2}-b^{2} c^{2}-2 a b c d-a^{2} d^{2}\)
\(=\left\{(a c)^{2}-2 \cdot a c \cdot b d+(b d)^{2}\right\}-\left(b^{2} c^{2}+2 a b c d+a^{2} d^{2}\right)\)
\(=(a c-b d)^{2}-(b c+a d)^{2}\)
\(=(a c-b d+b c+a d)(a c-b d-b c-a d)\)
\(=(a c+b c+a d-b d)(a c-b c-a d-b d)\)

\(a^{2}-b^{2}-4 a c+4 b c\)
\(=a^{2}-4 a c+4 c^{2}-b^{2}+4 b c-4 c^{2}\)
\(=\left\{(a)^{2}-2 \cdot a \cdot 2 c \cdot+(2 c)^{2}\right\}-\left\{(b)^{2}-2 \cdot b \cdot 2 c+(2 c)^{2}\right\}\)
\(=(a-2 c)^{2}-(b-2 c)^{2}\)
\(=(a-2 c+b-2 c)(a-2 c-b+2 c)\)
\(=(a+b-4 c)(a-b)\)
\(=(a-b)(a+b-4 c)\)

\(\left(a^{2}-b^{2}-c^{2}+d^{2}\right)^{2}-4(a d-b c)^{2}\)
\(=\left(a^{2}-b^{2}-c^{2}+d^{2}\right)^{2}-\{2(a d-b c)\}^{2}\)
\(=\left(a^{2}-b^{2}-c^{2}+d^{2}\right)^{2}-(2 a d-2 b c)^{2}\)
\(=\left(a^{2}-b^{2}-c^{2}+d^{2}+2 a d-2 b c\right)\)
\( \quad \quad \quad \quad \left(a^{2}-b^{2}-c^{2}+d^{2}-2 a d+2 b c\right)\)
\(=\{(a^{2}+2 a d+d^{2})-(b^{2}+2 b c+c^{2})\}\{ (a^{2}-2 a d+d^{2})\)
\(\quad \quad \quad \quad \quad \quad -\left(b^{2}-2 b c+c^{2}\right)\}\)
\(=\left\{(a+d)^{2}-(b+c)^{2}\right\}\left\{(a-d)^{2}-(b-c)^{2}\right\}\)
\(=\{(a+d+b+c)(a+d-b-c)\}\{(a-d+b-c)\)
\(\quad \quad \quad \quad (a-d-b+c)\}\)
\(=(a+b+c+d)(a-b-c-d)(a+b-c-d)(a-b+c-d)\)

\(3 x^{2}-y^{2}+z^{2}-2 x y-4 x z\)
\(=4 x^{2}-4 x z+z^{2}-x^{2}-2 x y-y^{2}\)
\(=\left\{(2 x)^{2}-2\cdot 2 x \cdot z+(z)^{2}\right\}-\left\{(x)^{2}+2 x y+(y)^{2}\right\}\)
\(=(2 x-z)^{2}-(x+y)^{2}\)
\(=(2 x-z+x+y)(2 x-z-x-y)\)
\(=(3 x+y-z)(x-y-z)\)

\(81 x^{4}+4 y^{4}\) \(=\left(9 x^{2}\right)^{2}+\left(2 y^{2}\right)^{2}\)
\(=\left(9 x^{2}+2 y^{2}\right)^{2}-2\cdot 9 x^{2} \cdot 2 y^{2}\)
\(=\left(9 x^{2}+2 y^{2}\right)^{2}-36 x^{2} y^{2}\)
\(=\left(9 x^{2}+2 y^{2}\right)^{2}-(6 x y)^{2}\)
\(=\left(9 x^{2}+2 y^{2}+6 x y\right)\left(9 x^{2}+2 y^{2}-6 x y\right)\)
\(=\left(9 x^{2}+6 x y+2 y^{2}\right)\left(9 x^{2}-6 x y+2 y^{2}\right)\)

\(\mathrm{p}^{4}-13 \mathrm{p}^{2} \mathrm{q}^{2}+4 \mathrm{q}^{4}\)
\(=\left\{\left(p^{2}\right)^{2}-2 \cdot p^{2} \cdot 2 q^{2}+\left(2 q^{2}\right)^{2}\right\}-9 p^{2} q^{2}\)
\(=\left(p^{2}-2 q^{2}\right)^{2}-(3 p q)^{2}\)
\(=\left(p^{2}-2 q^{2}+3 p q\right)\left(p^{2}-2 q^{2}-3 p q\right)\)
\(=\left(p^{2}+3 p q-2 q^{2}\right)\left(p^{2}-3 p q-2 q^{2}\right)\)

\(x^{8}-16 y^{8}\) \(=\left(x^{4}\right)^{2}-\left(4 y^{4}\right)^{2}\)
\(=\left(x^{4}+4 y^{4}\right)\left(x^{4}-4 y^{4}\right)\)
\(=\left(x^{4}+4 y^{4}\right)\left\{\left(x^{2}\right)^{2}-\left(2 y^{2}\right)^{2}\right\}\)
\(=\left\{\left(x^{2}\right)^{2}+\left(2 y^{2}\right)^{2}\right\}\left\{\left(x^{2}+2 y^{2}\right)\left(x^{2}-2 y^{2}\right)\right\}\)
\(=\left\{\left(x^{2}+2 y^{2}\right)^{2}-2 \cdot x^{2} \cdot 2 y^{2}\right\}\left\{\left(x^{2}+2 y^{2}\right)\left(x^{2}-2 y^{2}\right)\right\}\)
\(=\left\{\left(x^{2}+2 y^{2}\right)^{2}-4 x^{2} y^{2}\right\}\left\{\left(x^{2}+2 y^{2}\right)\left(x^{2}-2 y^{2}\right) \}\right.\)
\(=\left\{\left(x^{2}+2 y^{2}\right)^{2}-(2 x y)^{2}\right\}\left\{\left(x^{2}+2 y^{2}\right)\left(x^{2}-2 y^{2}\right)\right\}\)
\(=\left(x^{2}+2 y^{2}+2 x y\right)\left(x^{2}+2 y^{2}-2 x y\right)\left(x^{2}+2 y^{2}\right)\left(x^{2}-2 y^{2}\right)\)
\(=\left(x^{2}+2 x y+2 y^{2}\right)\left(x^{2}-2 x y+2 y^{2}\right)\left(x^{2}+2 y^{2}\right)\left(x^{2}-2 y^{2}\right)\)

\(x^{4}+x^{2} y^{2}+y^{4}\)
\(=\left(x^{2}\right)^{2}+2 x^{2} y^{2}+\left(y^{2}\right)^{2}-x^{2} y^{2}\)
\(=\left(x^{2}+y^{2}\right)^{2}-(x y)^{2}\)
\(=\left(x^{2}+y^{2}+x y\right)\left(x^{2}+y^{2}-x y\right)\)
\(=\left(x^{2}+x y+y^{2}\right)\left(x^{2}-x y+y^{2}\right)\)

\(3 x^{4}+2 x^{2} y^{2}-y^{4}\)
\(=4 x^{4}-x^{4}+2 x^{2} y^{2}-y^{4}\)
\(=\left(2 x^{2}\right)^{2}-\left(x^{4}-2 x^{2} y^{2}+y^{4}\right)\)
\(=\left(2 x^{2}\right)^{2}-\left\{\left(x^{2}-y^{2}\right)^{2}\right\}\)
\(=\left(2 x^{2}\right)^{2}-\left(x^{2}-y^{2}\right)^{2}\)
\(=\left(2 x^{2}+x^{2}-y^{2}\right)\left(2 x^{2}-x^{2}+y^{2}\right)\)
\(=\left(3 x^{2}-y^{2}\right)\left(x^{2}+y^{2}\right)\)
\(=\left(x^{2}+y^{2}\right)\left(3 x^{2}-y^{2}\right)\)

\(x^{4}+x^{2}+1\) \(=\left(x^{2}\right)^{2}+2 \cdot x^{2} \cdot 1+1^{2}-x^{2}\)
\(=\left(x^{2}+1\right)^{2}-(x)^{2}\)
\(=\left(x^{2}+1+x\right)\left(x^{2}+1-x\right)\)
\(=\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)\)

\(x^{4}+6 x^{2} y^{2}+8 y^{4}\)
\(=\left(x^{2}\right)^{2}+2 \cdot x^{2} \cdot 3 y^{2}+\left(3 y^{2}\right)^{2}-y^{4}\)
\(=\left(x^{2}+3 y^{2}\right)^{2}-\left(y^{2}\right)^{2}\)
\(=\left(x^{2}+3 y^{2}+y^{2}\right)\left(x^{2}+3 y^{2}-y^{2}\right)\)
\(=\left(x^{2}+4 y^{2}\right)\left(x^{2}+2 y^{2}\right)\)

\(3 x^{2}-y^{2}+z^{2}-2 x y-4 x z\)
\(=4 x^{2}-4 x z+z^{2}-x^{2}-2 x y-y^{2}\)
\(=(2 x)^{2}-2\cdot 2 x \cdot z+(z)^{2}-\left(x^{2}+2 x y+y^{2}\right)\)
\(=(2 x-z)^{2}-(x+y)^{2}\)
\(=(2 x-z+x+y)(2 x-z-x-y)\)
\(=(3 x+y-z)(x-y-z)\)

\(3 x^{4}-4 x^{2} y^{2}+y^{4}\)
\(=4 x^{4}-4 x^{2} y^{2}+y^{4}-x^{4}\)
\(=\left(2 x^{2}\right)^{2}-2 \cdot 2 x^{2} \cdot y^{2}+\left(y^{2}\right)^{2}-\left(x^{2}\right)^{2}\)
\(=\left(2 x^{2}-y^{2}\right)^{2}-\left(x^{2}\right)^{2}\)
\(=\left(2 x^{2}-y^{2}+x^{2}\right)\left(2 x^{2}-y^{2}-x^{2}\right)\)
\(=\left(3 x^{2}-y^{2}\right)\left(x^{2}-y^{2}\right)\)
\(=\left(3 x^{2}-y^{2}\right)(x+y)(x-y)\)
\(=(x+y)(x-y)\left(3 x^{2}-y^{2}\right)\)

\(p^{4}-2 p^{2} q^{2}-15 q^{4}\)
\(=\left(p^{2}\right)^{2}-2 \cdot p^{2} \cdot q^{2}+\left(q^{2}\right)^{2}-\left(q^{2}\right)^{2}-15 q^{4}\)
\(=\left(p^{2}-q^{2}\right)^{2}-16 q^{4}\)
\(=\left(p^{2}-q^{2}\right)^{2}-\left(4 q^{2}\right)^{2}\)
\(=\left(p^{2}-q^{2}+4 q^{2}\right)\left(p^{2}-q^{2}-4 q^{2}\right)\)
\(=\left(p^{2}+3 q^{2}\right)\left(p^{2}-5 q^{2}\right)\)

\(x^{8}+x^{4} y^{4}+y^{8}\)
\(=\left(x^{4}\right)^{2}+2 \cdot x^{4} \cdot y^{4}+\left(y^{4}\right)^{2}-x^{4} y^{4}\)
\(=\left(x^{4}+y^{4}\right)^{2}-\left(x^{2} y^{2}\right)^{2}\)
\(=\left(x^{4}+y^{4}+x^{2} y^{2}\right)\left(x^{4}+y^{4}-x^{2} y^{2}\right)\)
\(=\left\{\left(x^{2}\right)^{2}+2 \cdot x^{2} \cdot y^{2}+\left(y^{2}\right)^{2}-x^{2} y^{2}\right\}\left(x^{4}+y^{4}-x^{2} y^{2}\right)\)
\(=\left\{\left(x^{2}+y^{2}\right)^{2}-(x y)^{2}\right\}\left(x^{4}-x^{2} y^{2}+y^{4}\right)\)
\(=\left(x^{2}+y^{2}+x y\right)\left(x^{2}+y^{2}-x y\right)\left(x^{4}-x^{2} y^{2}+y^{4}\right)\)
\(=\left(x^{2}+x y+y^{2}\right)\left(x^{2}-x y+y^{2}\right)\left(x^{4}-x^{2} y^{2}+y^{4}\right)\)