Let $p$ and $q$ be real numbers such that the roots of
\[x^2 + px + q = 0\]
are real. Prove that the roots of
\[x^2 + px + q + (x + a)(2x + p) = 0\] are real, for any real number $a$
You know that x^2 + px + q = 0, so you just have to show that (x + a)(2x + p) = 0.
