If f x = 2 x 3 + m x 2 -   13 x + n and 2 ,   3 are roots of the equation f x = 0 , then the…

$f\left(x\right)=2x^3+{mx}^2- 13x+n$

Since $2 \& 3$ are roots of equation $f\left(x\right)=0$

So $f\left(2\right)=0$

$\Rightarrow 16+4m-26+n=0$

$\Rightarrow 4m+n-10=0 ....\left(\mathrm{i}\right)$

$f\left(3\right)=0$

$\Rightarrow 54+9m - 39+n=0$

$\Rightarrow 9m+n+15=0 ....\left(\mathrm{ii}\right)$

Subtracting equation $\left(\mathrm{i}\right)$ and $\left(\mathrm{ii}\right)$, we get

$5m+25=0$

$\Rightarrow m=- 5$

Put $m=-5$ in equation $\left(\mathrm{i}\right)$, we get 

$n=30$