I need help trying to put the correct vocabulary words when solving this problem such as grouping, factor, prime factor, GCF, and perfect square, and please check and see if I am doing this correctly.

15x^2+31x+2

In my last problem I will be using the “ac method”

A*C = 15*2 = 30 = 1*30. Sum = 1+30=31=B.

Ac=30

Factors must add to 31

30+1 = 31

(x+2)(15x+1)

15x^2 + (x+30x) + 2

(15x^2+x) + (30x+2) =

X (15x+1) + 2(15x+1) =

15x^2 + 30x + x + 2

15x(x+2) + 15x+2

(15x+1)(x+2)

Final answer: 15x(x+2) + 1(x+2)

15x(x+2) + 1(x+2)

Factor out x+2:
(x+2)(15x+1) = Final answer.

It looks like you are using the "ac method" to factor the quadratic expression 15x^2 + 31x + 2 correctly. Here's a breakdown of your steps:

1. Identify the values for A, B, and C in the quadratic expression: A = 15, B = 31, C = 2.
2. Use the "ac method" to find two numbers that multiply to give AC (A*C) and add up to B. In this case, AC = 15*2 = 30, and you correctly found that 1 and 30 satisfy the condition (1 + 30 = 31 = B).
3. Express the middle term (31x) using the two numbers found in step 2. Rewrite the expression as: 15x^2 + (x + 30x) + 2.
4. Group the terms in pairs: (15x^2 + x) + (30x + 2).
5. Factor out the common factors from each pair: x(15x + 1) + 2(15x + 1).
6. Combine the like terms within each pair: 15x(x + 2) + 1(x + 2).
7. Finally, factor out the common binomial factor (15x + 1) to obtain the final answer: (15x + 1)(x + 2).

Overall, your work is correct and you have used the vocabulary words correctly. The terms "grouping" refers to the step where you group the terms into pairs, "factor" refers to extracting common factors from terms, "prime factor" is not directly used in your work, "GCF" stands for Greatest Common Factor which is used to extract common factors, and "perfect square" does not apply to the current problem. Good job!