How would you factor 4x^2 + 9? i think the answer has imaginary numbers in it, but how would you get the solution?

The answer is it cannot be. It is because if you use the difference of two square, the operation used must not be addition, but it would be subtraction.
It can be factored out if you will find the factors of 4x^2 - 9 =0
So the factors will be(using difference of two perfect squares):
(2x+3)(2x-3)
Cause if you will use the factors (2x+3)(2x+3) or even the negative signs to it, it will give:
4x^2+12x+9 or 4x^2-12x-9 

so for the answer of your question, it is invalid.