Verify division algorithm for the polynomial $f(x)=\left(8+20 x+x^{2}-6 x^{3}\right)$ by $g(x)=(2+5 x-$ $\left.3 x^{2}\right)$

$-6 x^{3}+x^{2}+20 x+8$ and $g(x)$ as $-3 x^{2}+5 x+2$

Quotient $=2 \mathrm{x}+3$

Remainder $=x+2$

By using division rule, we have

Dividend $=$ Quotient $\times$ Divisor $+$ Remainder

$\therefore-6 x^{3}+x^{2}+20 x+8=\left(-3 x^{2}+5 x+2\right)(2 x+3)+x+2$

$\Rightarrow-6 x^{3}+x^{2}+20 x+8=-6 x^{3}+10 x^{2}+4 x-9 x^{2}+15 x+6+x+2$

$\Rightarrow-6 \mathrm{x}^{3}+\mathrm{x}^{2}+20 \mathrm{x}+8=-6 \mathrm{x}^{3}+\mathrm{x}^{2}+20 \mathrm{x}+8$