Question: Factor the trinomial. \[ a^2 - 4ag - 32g^2 \] Select the correct choice below and, if…

Factor the trinomial.

\[ a^2 - 4ag - 32g^2 \]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. \( a^2 - 4ag - 32g^2 = \boxed{\phantom{Factor completely.}} \) (Factor completely.)

B. The polynomial is prime.

Solution

To factor the trinomial \( a^2 - 4ag - 32g^2 \), we need to find two numbers that multiply to the product of the coefficient of \( a^2 \) (which is 1) and the constant term (-32) and add up to the coefficient of the middle term (-4). The two numbers are 4 and -8. Now, let’s rewrite the middle term using these numbers: \[ a^2 - 4ag - 32g^2 = a^2 + 4ag - 8ag - 32g^2 \] Next, group the terms and factor by grouping: \[ (a^2 + 4ag) + (-8ag - 32g^2) \] Factor out the greatest common factor from each group: \[ a(a + 4g) - 8g(a + 4g) \] Notice that \(a + 4g\) is a common factor: \[ (a - 8g)(a + 4g) \] Thus, the factored form of the trinomial is \( (a - 8g)(a + 4g) \).