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 $ax^{2}+bx+c=0$.
$$-2x^{2}+3x+4=0$$
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 $x=\frac{-3±\sqrt{41}}{-4}$ when $±$ is plus. Add $-3$ to $\sqrt{41}$.
$$x=\frac{\sqrt{41}-3}{-4}$$
Divide $-3+\sqrt{41}$ by $-4$.
$$x=\frac{3-\sqrt{41}}{4}$$
Now solve the equation $x=\frac{-3±\sqrt{41}}{-4}$ when $±$ is minus. Subtract $\sqrt{41}$ from $-3$.
$$x=\frac{-\sqrt{41}-3}{-4}$$
Divide $-3-\sqrt{41}$ by $-4$.
$$x=\frac{\sqrt{41}+3}{4}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{3-\sqrt{41}}{4}$ for $x_{1}$ and $\frac{3+\sqrt{41}}{4}$ for $x_{2}$.
