Yours Truly answers: Textbooks would say that we should make a list of factors of  4  and  9, and “try them.” In reality, this process frustrates us because we feel as if we depend on luck.
 
Here is how we can more efficiently find the correct factorization.
Step 1.  4 x 9 = 36 which is nowhere close to 15.
Step 2.  Leave  4  as is and break  9  into  3·3. Then, we see that  (4 x 3) + (1 x 3) = 15.
Step 3.  So, the factorization is  4x² + 15x + 9 = (4x + 3)(x + 3).