For how many integer values of $a$ does the equation $x^2 + ax + 5a = 0$ have integer solutions for $x$?

Here's what I find.....

Let a = m....the discriminant is

m^2 -  4*5 * (m)  = m^2 - 20m  = m(m - 20)

This will be a perfect square when m = 20   ⇒ a = 20

And

x^2  + 20x +5(20) =

x^2 + 20x + 100       has  the root (-10, 0) 

And if a = 0.....this has the root   (0, 0)  

So

a = 0   or  a = 20