Answer:
\[ = 4(3x^2 + 8x - 3) \]
Step-by-step explanation:
Part A: The greatest common factor (GCF) of the given expression 12x^2 + 32x - 12 is 4. To find the GCF, identify the common factors of the coefficients (12, 32, and 12) and the variable terms (x^2 and x) and take the smallest power of each.
Part B: Factoring the expression completely:
\[ 12x^2 + 32x - 12 \]
\[ = 4(3x^2 + 8x - 3) \]
Now, we need to factor the quadratic expression \(3x^2 + 8x - 3\). Since this quadratic doesn't factor easily, you can use the quadratic formula or factoring by grouping to find the two binomials.
The factored form is:
\[ 4(3x - 1)(x + 3) \]
Part C: Checking the factoring by multiplying:
\[ 4(3x - 1)(x + 3) \]
\[ = 4(3x^2 + 9x - x - 3) \]
\[ = 4(3x^2 + 8x - 3) \]
This result matches the original expression \(12x^2 + 32x - 12\), confirming that the factoring is correct.
