find all integer numbers a for which 4a^2-24a-45 is a prime number

Since that quadratic factors as (2a+3)(2a-15), the only way it can be prime is if one of the factors is 1 with the other factor being prime, or -1 with the other factor being a negative prime.

Case 1… 2a+3=1 giving a=-1 and the other factor -17… NG

Case 2 … 2a+3=-1 giving a=-2 and the other factor -19 … making the quadratic = 19 👍

Case 3 … 2a-15=1 giving a=8 and the other factor 19 … making the quadratic = 19 👍

Case 4 … 2a-15=-1 giving a=7 and the other factor 17 … NG.

Hence, only a=-2 and a=8.