I think you mean a^4+20a^2-576 as the expression.
This is a quadratic in a^2: (a^2)^2+20(a^2)-576, treating a^2 like a single variable.
We're looking for factors of 576 whose difference is 20. The factors are 16 and 36.
This takes us to the next stage: (a^2-16)(a^2+36),
If we for the moment discount complex numbers, we can factorise further:
(a-4)(a+4)(a^2+36). This gives us two zeroes, 4 and -4.
The other two are complex: (a-6i)(a+6i) where i=√-1. So a=6i and -6i.
Complete factorisation is (a-4)(a+4)(a-6i)(a+6i).