I'm stuck on this homework problem. Can someone please give me a hint?
Given $2a^2+3ab+4ac<0$. Is the following expression zero, negative or positive? $$9b^2-32ac$$
Here's my suggestion: Consider the quadratic function $f(x) = 2a^2 x^2 + 3abx + 4ac$. Then your initial inequality can be expressed as $f(1) < 0$, and the discriminant of $f(x)$ is given by $a^2(9b^2 - 32ac)$. Now investigate the behavior of the roots of $f(x)$. It might be useful to observe that $f(x)$ is a "happy" parabola.