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 $ax^{2}+bx+c=0$.
$$2x^{2}-4x-47=0$$
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 $x=\frac{4±14\sqrt{2}}{4}$ when $±$ is plus. Add $4$ to $14\sqrt{2}$.
$$x=\frac{14\sqrt{2}+4}{4}$$
Divide $4+14\sqrt{2}$ by $4$.
$$x=\frac{7\sqrt{2}}{2}+1$$
Now solve the equation $x=\frac{4±14\sqrt{2}}{4}$ when $±$ is minus. Subtract $14\sqrt{2}$ from $4$.
$$x=\frac{4-14\sqrt{2}}{4}$$
Divide $4-14\sqrt{2}$ by $4$.
$$x=-\frac{7\sqrt{2}}{2}+1$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $1+\frac{7\sqrt{2}}{2}$ for $x_{1}$ and $1-\frac{7\sqrt{2}}{2}$ for $x_{2}$.
