All the roots of \(x^2 + px + q = 0\) are real, where \(p\) and \(q\) are real numbers. Prove that all 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 = 0 are real, and the roots of (x + a)(2x + p) = 0 are real, so the roots of x^2 + px + q + (x + a)(2x + p) = 0 are real.