Factorising polynomials 2

Question

Hi I need help solving this question please, I'am in year 10:

If $(x − p)$ is a factor of $mx^2 + nx + q$, show that $−2\sqrt{mq} \le n \le 2\sqrt{mq}$.

Answer 1

If $x - p$ is a factor of said quadratic, where $p \in \mathbb{R}$, then the equation

$$mx^2 + nx + q = 0$$

has at least one real root. From the chapter of discriminants, this means that the discriminant of said equation is either $0$ or positive. That is,

$$n^2 - 4mq \ge 0$$ $$n^2 \ge 4mq$$

Now go back to the chapter on inequalities to find the range of $n$ in terms of $m$ and $q$ for which this inequality is satisfied.