\(2 a^{2}+5 a+2\)
\( =2 a^{2}+(4+1) a+2\)
\(=2 a^{2}+4 a+a+2\)
\(=2 a(a+2)+1(a+2)\)
\(=(a+2)(2 a+1) \)

\(3 x^{2}+14 x+8\)
\( =3 x^{2}+(12+2) x+8\)
\(=3 x^{2}+12 x+2 x+8\)
\(=3 x(x+4)+2(x+4)\)
\(=(x+4)(3 x+2) \)

\(2 m^{2}+7 m+6\)
\( =2 m^{2}+(4+3) m+6\)
\(=2 m^{2}+4 m+3 m+6\)
\(=2 m(m+2)+3(m+2)\)
\(=(m+2)(2 m+3) \)

\(6 x^{2}-x-15\)
\( =6 x^{2}-(10-9) x-15\)
\(=6 x^{2}-10 x+9 x-15\)
\(=2 x(3 x-5)+3(3 x-5)\)
\(=(3 x-5)(2 x+3) \)

\(9 r^{2}+r-8\)
\( =9 r^{2}+(9-8) r-8\)
\(=9 r^{2}+9 r-8 r-8\)
\(=9 r(r+1)-8(r+1)\)
\(=(r+1)(9 r-8) \)

\(6 m^{2}-11 m n-10 n^{2}\)
\( =6 m^{2}-(15-4) m n-10 n^{2}\)
\(=6 m^{2}-15 m n+4 m n-10 n^{2}\)
\(=3 m(2 m-5 n)+2 n(2 m-5 n)\)
\(=(2 m-5 n)(3 m+2 n) \)

\(7 x^{2}+48 x y-7 y^{2}\)
\( =7 x^{2}+(49-1) x y-7 y^{2}\)
\(=7 x^{2}+49 x y-x y-7 y^{2}\)
\(=7 x(x+7 y)-y(x+7 y)\)
\(=(x+7 y)(7 x-y) \)

\(12+x-6 x^{2}\)
\( =12+(9-8) x-6 x^{2}\)
\(=12+9 x-8 x-6 x^{2}\)
\(=3(4+3 x)-2 x(4+3 x)\)
\(=(4+3 x)(3-2 x) \)

\(6+5 a-6 a^{2}\)
\( =6+(9-4) a-6 a^{2}\)
\(=6+9 a-4 a-6 a^{2}\)
\(=3(2+3 a)-2 a(2+3 a)\)
\(=(2+3 a)(3-2 a) \)

\(6 x^{2}-13 x+6\)
\( =6 x^{2}-(9+4) x+6\)
\(=6 x^{2}-9 x-4 x+6\)
\(=3 x(2 x-3)-2(2 x-3)\)
\(=(2 x-3)(3 x-2) \)

\(99 a^{2}-202 a b+99 b^{2}\)
\( =99 a^{2}-(121+81) a b+99 b^{2}\)
\(=99 a^{2}-121 a b-81 a b+99 b^{2}\)
\(=11 a(9 a-11 b)-9 b(9 a-11 b)\)
\(=(9 a-11 b)(11 a-9 b) \)
বিকল্প পদ্ধতি :
\(99 a^{2}-202 a b+99 b^{2}\)
\( =(100-1) a^{2}-(200+2) a b+(100-1) b^{2}\)
\(=\left(100 a^{2}-200 a b+100 b^{2}\right)-\left(a^{2}+2 a b+b^{2}\right)\)
\(=\left\{(10 a)^{2}-2 \cdot 10 a \cdot 10 b+(10 b)^{2}\right\}-(a+b)^{2}\)
\(=(10 a-10 b)^{2}-(a+b)^{2}\)
\(=(10 a-10 b+a+b)(10 a-10 b-a-b)\)
\(=(11 a-9 b)(9 a-11 b) \)

\(2 a^{6}-13 a^{3}-24\)
\( =2 a^{6}-(16-3) a^{3}-24\)
\(=2 a^{6}-16 a^{3}+3 a^{3}-24\)
\(=2 a^{3}\left(a^{3}-8\right)+3\left(a^{3}-8\right)\)
\(=\left(a^{3}-8\right)\left(2 a^{3}+3\right)\)
\(=\left\{(a)^{3}-(2)^{3}\right\}\left(2 a^{3}+3\right)\)
\(=(a-2)\left(a^{2}+2 a+4\right)\left(2 a^{3}+3\right) \)

\(8 a^{4}+2 a^{2}-45\)
\( =8 a^{4}+(20-18) a^{2}-45\)
\(=8 a^{4}+20 a^{2}-18 a^{2}-45\)
\(=4 a^{2}\left(2 a^{2}+5\right)-9\left(2 a^{2}+5\right)\)
\(=\left(2 a^{2}+5\right)\left(4 a^{2}-9\right)\)
\(=\left(2 a^{2}+5\right)\left\{(2 a)^{2}-(3)^{2}\right\}\)
\(=\left(2 a^{2}+5\right)(2 a+3)(2 a-3) \)

\(6(x-y)^{2}-x+y-15\)
\( =6(x-y)^{2}-(x-y)-15\)
\(=6 p^{2}-p-15 \) [ধরি, \( x-y=p \)]
\(=6 p^{2}-(10-9) p-15\)
\(=6 p^{2}-10 p+9 p-15\)
\(=2 p(3 p-5)+3(3 p-5)\)
\(=(3 p-5)(2 p+3)\)
\(=\{3(x-y)-5\}\{2(x-y)+3\}\) [\(p\)-এর মান বসিয়ে]
\(=(3 x-3 y-5)(2 x-2 y+3) \)

\(3(a+b)^{2}-2 a-2 b-8\)
\( =3(a+b)^{2}-2(a+b)-8\)
\(=3 p^{2}-2 p-8\) [ধরি, \(a+b=p \) ]
\(=3 p^{2}-(6-4) p-8\)
\(=3 p^{2}-6 p+4 p-8\)
\(=3 p(p-2)+4(p-2)\)
\(=(p-2)(3 p+4)\)
\(=\{(a+b)-2\}\{3(a+b)+4\}\) [ \(p\)-এর মান বসিয়ে]
\(=(a+b-2)(3 a+3 b+4) \)

\(6(a+b)^{2}+5\left(a^{2}-b^{2}\right)-6(a-b)^{2}\)
\(=6(a+b)^{2}+5(a+b)(a-b)-6(a-b)^{2}\)
ধরি, \(a+b=x\)
\(a-b=y\)
\(=6 x^{2}+5 x y-6 y^{2}\)
\(=6 x^{2}+9 x y-4 x y-6 y^{2}\)
\(=3 x(2 x+3 y)-2 y(2 x+3 y)\)
\(=(2 x+3 y)(3 x-2 y)\)
\(=\{2(a+b)+3(a-b)\}\{3(a+b)-2(a-b)\}\)
\(=(2 a+2 b+3 a-3 b)(3 a+3 b-2 a+2 b)\)
\(=(5 a-b)(a+5b)\)

\(x^{2}-2 x-3\)
\(=x^{2}-2 \cdot x+1-4\)
\(=(x-1)^{2}-(2)^{2}\)
\(=(x-1+2)(x-1-2)\)
\(=(x+1)(x-3)\)

\(x^{2}+5 x+6\)
\( =x^{2}+2 \cdot x \cdot \frac{5}{2}+\left(\frac{5}{2}\right)^{2}-\left(\frac{5}{2}\right)^{2}+6\)
\(=\left(x+\frac{5}{2}\right)^{2}-\frac{25}{4}+6\)
\(=\left(x+\frac{5}{2}\right)^{2}-\left(\frac{25}{4}-6\right)\)
\(=\left(x+\frac{5}{2}\right)^{2}-\left(\frac{25-24}{4}\right)\)
\(=\left(x+\frac{5}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2} \)
\( =\left(x+\frac{5}{2}+\frac{1}{2}\right)\left(x+\frac{5}{2}-\frac{1}{2}\right)\)
\(=\left(\frac{2 x+5+1}{2}\right)\left(\frac{2 x+5-1}{2}\right)\)
\(=\left(\frac{2 x+6}{2}\right)\left(\frac{2 x+4}{2}\right)\)
\(=\frac{2(x+3)}{2} \times \frac{2(x+2)}{2}\)
\(=(x+3)(x+2) \)

\(3 x^{2}-7 x-6\)
\( =3\left[x^{2}-\frac{7}{3} x-2\right]\)
\(=3\left[x^{2}-2 \cdot x \cdot \frac{7}{6}+\left(\frac{7}{6}\right)^{2}-\frac{49}{36}-2\right]\)
\(=3\left[\left(x-\frac{7}{6}\right)^{2}-\left(\frac{49}{36}+2\right)\right]\)
\(=3\left[\left(x-\frac{7}{6}\right)^{2}-\left(\frac{49+72}{36}\right)\right]\)
\(=3\left[\left(x-\frac{7}{6}\right)^{2}-\frac{121}{36}\right]\)
\(=3\left[\left(x-\frac{7}{6}\right)^{2}-\left(\frac{11}{6}\right)^{2}\right]\)
\(=3\left(x-\frac{7}{6}+\frac{11}{6}\right)\left(x-\frac{7}{6}-\frac{11}{6}\right)\)
\(=3\left(\frac{6 x-7+11}{6}\right)\left(\frac{6 x-7-11}{6}\right)\)
\(=3\left(\frac{6 x+4}{6}\right)\left(\frac{6 x-18}{6}\right)\)
\(=3\left(x+\frac{2}{3}\right)(x-3)=3\left(\frac{3 x+2}{3}\right)(x-3)\)
\(=(3 x+2)(x-3) \)

\(3 a^{2}-2 a-5\)
\( =3\left[a^{2}-\frac{2}{3} \cdot a-\frac{5}{3}\right]\)
\(=3\left[a^{2}-2 \cdot a \cdot \frac{1}{3}+\left(\frac{1}{3}\right)^{2}-\frac{1}{9}-\frac{5}{3}\right]\)
\(=3\left[\left(a-\frac{1}{3}\right)^{2}-\left(\frac{1}{9}+\frac{5}{3}\right)\right]\)
\(=3\left[\left(a-\frac{1}{3}\right)^{2}-\left(\frac{1+15}{9}\right)\right]\)
\(=3\left[\left(a-\frac{1}{3}\right)^{2}-\frac{16}{9}\right]\)
\(=3\left[\left(a-\frac{1}{3}\right)^{2}-\left(\frac{4}{3}\right)^{2}\right]\)
\(=3\left(a-\frac{1}{3}+\frac{4}{3}\right)\left(a-\frac{1}{3}-\frac{4}{3}\right)\)
\(=3\left(\frac{3 a-1+4}{3}\right)\left(\frac{3 a-1-4}{3}\right)\)
\(=3(a+1)\left(\frac{3 a-5}{3}\right)\)
\(=(a+1)(3 a-5) \)

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

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

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

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

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

\(a^{2}+1-\frac{6}{a^{2}}\)
\( =\frac{1}{a^{2}}\left[a^{4}+a^{2}-6\right]\)
\(=\frac{1}{a^{2}}\left[a^{4}+3 a^{2}-2 a^{2}-6\right]\)
\(=\frac{1}{a^{2}}\left[a^{2}\left(a^{2}+3\right)-2\left(a^{2}+3\right)\right]\)
\(=\frac{1}{a^{2}}\left[\left(a^{2}+3\right)\left(a^{2}-2\right)\right]\)
\(=\frac{\left(a^{2}+3\right)}{a} \times \frac{\left(a^{2}-2\right)}{a}\)
\(=\left(a+\frac{3}{a}\right)\left(a-\frac{2}{a}\right) \)
বিকল্প পদ্ধতি :
\( a^{2}+1-\frac{6}{a^{2}}=a^{2}+(3-2) \frac{a}{a}-\frac{6}{a^{2}} \)
\( =a^{2}+\frac{3 a}{a}-\frac{2 a}{a}-\frac{6}{a^{2}}\)
\(=a\left(a+\frac{3}{a}\right)-\frac{2}{a}\left(a+\frac{3}{a}\right)\)
\(=\left(a+\frac{3}{a}\right)\left(a-\frac{2}{a}\right) \)