Final answer:
To make the trinomial x^2 + 6x + 32 factorable, the 'b' value should be changed to 12, making the trinomial factorable into (x+4)(x+8), as 4 and 8 are factors of 32 that add up to 12.
Step-by-step explanation:
The trinomial x^2 + 6x + 32 is indeed prime with the current 'b' value of 6. To make this trinomial factorable, we need to modify the 'b' value. A common strategy for factoring is to look for two numbers that multiply to the 'c' term (which is 32 in this case) while adding up to the 'b' value.
When factoring a quadratic trinomial of the form ax^2 + bx + c, we look for a pair of numbers that multiply to a*c and add up to b. In this scenario, because 'a' is 1, we are just looking for factors of 32 that add up to a new 'b' value. The original 'b' value of 6 does not meet this requirement, because there are no two integers that can multiply to 32 and add up to 6.
However, if we consider the factors of 32, they are 1, 2, 4, 8, 16, and 32. The pair among these factors that adds up to 8 is 4 and 8 since 4*8 = 32 and 4+8 = 12. Therefore, changing the 'b' value to 12 will make the trinomial factorable into (x+4)(x+8). This is an application of the properties of integer powers and simplifying algebra.