Knowing that 2 and 3 are the roots of the equation $2{\mathrm{x}}^3+{\mathrm{mx}}^2-13\mathrm{x}+\mathrm{n}=0$, determine m + n +2(p)? Where p is the root of the given equation.

Given the equation,
$2{\mathrm{x}}^3+{\mathrm{mx}}^2-13\mathrm{x}+\mathrm{n}=0$
Roots of the equation = 2 and 3
So, put x=2 and x=3 in the given polynomial.
Given polynomial,
$2{\mathrm{x}}^2+\mathrm{mx}-13\mathrm{x}+\mathrm{n}$
At x=2 and equate to zero,
$\mathrm{f}\left(2\right)=2(2{)}^3+\mathrm{m}(2{)}^2-13\left(2\right)+\mathrm{n}=0\mathrm{ }$
$\Rightarrow 16+4\mathrm{m}-26+\mathrm{n}=0\mathrm{ }$
$\Rightarrow 4\mathrm{m}+\mathrm{n}=10\dots \dots \left(1\right)\mathrm{ }$ 

At, x= 3 and equate to zero,
$\mathrm{f}\left(3\right)=2(3{)}^3+\mathrm{m}(3{)}^2-13\left(3\right)+\mathrm{n}=0\mathrm{ }$
$\Rightarrow 54+9\mathrm{m}-39+\mathrm{n}=0\mathrm{ }$
$\Rightarrow 9\mathrm{m}+\mathrm{n}=-15\dots \dots .\left(2\right)\mathrm{ }$

Subtract equation (1) from (2):
$9\mathrm{m}+\mathrm{n}-4 \mathrm{m}-\mathrm{n}=-15-10\mathrm{ }$
$5\mathrm{m}=-25\mathrm{ }$
$\mathbf{m}\mathbf{=}\frac{\mathbf{25}}{\mathbf{-}\mathbf{5}}\mathbf{=}\mathbf{-}\mathbf{5}\mathbf{ }$
$\mathrm{m}=-5\mathrm{ }$  Now we put the value of m in equation (1)
$\Rightarrow 4\mathrm{m}+\mathrm{n}=10\mathrm{ }$
$\Rightarrow 4\left(-5\right)+\mathrm{n}=10\mathrm{ }$
$\Rightarrow -20+\mathrm{n}=10\mathrm{ }$
$\Rightarrow \mathbf{n}\mathbf{=}\mathbf{30}\mathbf{ }$

Therefore, the value of $\mathrm{m}=-5$ and $\mathrm{n}=30$.

The sum of cubic roots,
$\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }=-\frac{\mathrm{b}}{\mathrm{a}}$
$2+3+\mathrm{p}=\frac{5}{2}\mathrm{ }$
$5+\mathrm{p}=\frac{5}{2}\mathrm{ }$
$\mathrm{p}=\frac{5}{2}-\frac{5}{1}\mathrm{ }$
$\mathrm{p}=\frac{5-10}{2}\mathrm{ }$
$\mathrm{p}=\frac{-5}{2}\mathrm{ }$                   [third root]
As per given question we have determine $\mathrm{m}+\mathrm{n}+2\left(\mathrm{p}\right)$ then,
$\mathrm{m}+\mathrm{n}+2\left(\mathrm{p}\right)=-5+30+2\left(-\frac{5}{2}\right)\mathrm{ }$
$=25-5\mathrm{ }$
$=20$
Hence, the required value of $\mathrm{m}+\mathrm{n}+2\left(\mathrm{p}\right)=20$