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 $a.$

Heather is writing a quadratic function that represents a parabola that touches but does not cross the x-axis at x = -6. which function could heather be writing? fx) = x2 + 36x + 12 = x2 - 36x - 12 f(x) = -x + 12x + 36 f(x) = -x? - 12x - 36