Answer
The required solution is $\left( x-\frac{1}{2} \right)\left( \text{ }x+2 \right)\left( x-3 \right)$.
Work Step by Step
The given relation shows that when the equation $2{{x}^{3}}-3{{x}^{2}}-11x-6$ is divided by $x-3$ , the remainder is zero and the quotient is $2{{x}^{2}}+3x-2$. So, $x-3$ is a solution to the given cubic equation or 3 is one of the roots of the equation. Now, we have to find out the remaining two roots of the equation; solve the quotient equation as follows:
$\begin{align}
& 2{{x}^{2}}+3x-2=0 \\
& 2{{x}^{2}}+4x-x-2=0 \\
& 2x\left( x+2 \right)-1\left( x+2 \right)=0 \\
& \left( 2x-1 \right)\left( x+2 \right)=0
\end{align}$
$x=\frac{1}{2}\text{ or }x=-2$
Thus, the remaining two solutions to the given cubic equation are $x-\frac{1}{2}\text{ and }x+2$.
