On dividing $3 x^{3}+x^{2}+2 x+5$ is divided by a polynomial $g(x)$, the quotient and remainder are $(3 x-5)$ and $(9 x+10)$ respectively. Find $g(x)$

using division rule,

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

$\therefore 3 x^{3}+x^{2}+2 x+5=(3 x-5) g(x)+9 x+10$

$\Rightarrow 3 x^{3}+x^{2}+2 x+5-9 x-10=(3 x-5) g(x)$

$\Rightarrow 3 x^{3}+x^{2}-7 x-5=(3 x-5) g(x)$

$\Rightarrow g(x)=\frac{3 x^{3}+x^{2}-7 x-5}{3 x-5}$

$3 x-5 \quad \frac{x^{2}+2 x+1}{3 x^{3}+x^{2}-7 x-5}{3 x^{3}-5 x^{2}}$

$6 x^{2}-10 x$

$3 x-5$