How do you factor $2{{x}^{2}}+3x+6$?

Hint: We will find the roots of the expression by completing the square method. First we will divide the expression by a. Now we will add and subtract the term ${{\left( \dfrac{b}{2a} \right)}^{2}}$ on both sides. Now simplify the expression by using the formula ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$ . Now we will rearrange the terms and take square roots on both sides and hence find the roots of the expression. Now the factors of the expression are nothing but $x-roots$ . Hence we will use this to find the factors of the expression.

Complete step by step solution:
Now we are given with a quadratic expression $2{{x}^{2}}+3x+6$ of the form $a{{x}^{2}}+bx+c$ .
Now first we will find the roots of the expression. To find the roots of the expression we will use the completing the square method.
Now consider the equation $2{{x}^{2}}+3x+6=0$
First we will divide the whole equation by 2, Hence we get, ${{x}^{2}}+\dfrac{3}{2}x+3=0$
Now we will add and subtract ${{\left( \dfrac{b}{2a} \right)}^{2}}$
Hence we get the equation as,
$\begin{align}
  & \Rightarrow {{x}^{2}}+\dfrac{3}{2}x+3+\dfrac{9}{16}-\dfrac{9}{16}=0 \\
 & \Rightarrow {{x}^{2}}+\dfrac{3}{2}x+\dfrac{9}{16}+\left( 3-\dfrac{9}{16} \right)=0 \\
\end{align}$
Now using the formula ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$ in the above expression we get,
$\Rightarrow {{\left( x+\dfrac{3}{4} \right)}^{2}}+\left( \dfrac{16\times 3-9}{16} \right)=0$
$\Rightarrow {{\left( x+\dfrac{3}{4} \right)}^{2}}=-\dfrac{39}{16}$
Now taking square root on both sides we get,
$\Rightarrow x+\dfrac{3}{4}=\dfrac{\pm \sqrt{39}i}{4}$
Now rearranging the terms in the above expression we get,
$\Rightarrow x=\dfrac{3\pm \sqrt{39}i}{4}$
Hence we have the roots of the given expression are, $\left( \dfrac{3-\sqrt{39}i}{4} \right)$ and $\left( \dfrac{3+\sqrt{39}i}{4} \right)$
Now we know that if $\alpha $ and $\beta $ are the roots of the given expression then the factors of the expression are $\left( x-\alpha \right)$ and $\left( x-\beta \right)$ .
Hence the factors of the above expression are $x-\left( \dfrac{3-\sqrt{39}i}{4} \right)$ and $x-\left( \dfrac{3+\sqrt{39}i}{4} \right)$ .

Note: Now note that we can also directly find the roots of the quadratic expression. The roots of the equation of the form $a{{x}^{2}}+bx+c$ is given by the formula $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ . Hence we can easily find the roots using this formula and hence also find the required factors of the expression.