Quadratic polynomial can be factored using the transformation $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$, where $x_{1}$ and $x_{2}$ are the solutions of the quadratic equation $ax^{2}+bx+c=0$.
$$-2a^{2}-4a+3=0$$
All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. The quadratic formula gives two solutions, one when $±$ is addition and one when it is subtraction.
Now solve the equation $a=\frac{4±2\sqrt{10}}{-4}$ when $±$ is plus. Add $4$ to $2\sqrt{10}$.
$$a=\frac{2\sqrt{10}+4}{-4}$$
Divide $4+2\sqrt{10}$ by $-4$.
$$a=-\frac{\sqrt{10}}{2}-1$$
Now solve the equation $a=\frac{4±2\sqrt{10}}{-4}$ when $±$ is minus. Subtract $2\sqrt{10}$ from $4$.
$$a=\frac{4-2\sqrt{10}}{-4}$$
Divide $4-2\sqrt{10}$ by $-4$.
$$a=\frac{\sqrt{10}}{2}-1$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $-1-\frac{\sqrt{10}}{2}$ for $x_{1}$ and $-1+\frac{\sqrt{10}}{2}$ for $x_{2}$.
