The roots of \[x^2 + px + q = 0\]are real, where $p$ and $q$ are real numbers. Prove that the roots of \[x^2 + px + q + (x + a)(2x + p) = 0\]are real, for any real number $a.$

The roots of x^2 + px + q are real, and the roots of (x + a)(2x + p) are real, so when we add them, we get another quadratic where the roots are real.