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 $ax^{2}+bx+c=0$.
$$5x^{2}+3x=0$$
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.
$$x=\frac{-3±\sqrt{3^{2}}}{2\times 5}$$
Take the square root of $3^{2}$.
$$x=\frac{-3±3}{2\times 5}$$
Multiply $2$ times $5$.
$$x=\frac{-3±3}{10}$$
Now solve the equation $x=\frac{-3±3}{10}$ when $±$ is plus. Add $-3$ to $3$.
$$x=\frac{0}{10}$$
Divide $0$ by $10$.
$$x=0$$
Now solve the equation $x=\frac{-3±3}{10}$ when $±$ is minus. Subtract $3$ from $-3$.
$$x=-\frac{6}{10}$$
Reduce the fraction $\frac{-6}{10}$ to lowest terms by extracting and canceling out $2$.
$$x=-\frac{3}{5}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $0$ for $x_{1}$ and $-\frac{3}{5}$ for $x_{2}$.
