Question: Factor the trinomial. \[ 20q^2 - 23qb + 6b^2 \] Select the correct choice below and, if…

Factor the trinomial.

\[ 20q^2 - 23qb + 6b^2 \]

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

A. \( 20q^2 - 23qb + 6b^2 = \boxed{\text{(Factor completely.)}} \)

B. The trinomial is prime.

Solution

To factor the trinomial \(20q^2 - 23qb + 6b^2\), we will follow these steps. First, identify the quadratic in standard form \(ax^2 + bx + c\) where: - \(a = 20\), - \(b = -23\), - \(c = 6\). We need to find two numbers that multiply to \(ac = 20 \times 6 = 120\) and add to \(-23\). The numbers are \(-15\) and \(-8\). Rewrite the middle term using these numbers: \[ 20q^2 - 15qb - 8qb + 6b^2 \] Group the terms: \[ (20q^2 - 15qb) + (-8qb + 6b^2) \] Factor each group: \[ 5q(4q - 3b) - 2b(4q - 3b) \] Notice that \(4q - 3b\) is a common factor: \[ (5q - 2b)(4q - 3b) \] Thus, the trinomial factors as \((5q - 2b)(4q - 3b)\). The correct answer is option A: \[ 20q^2 - 23qb + 6b^2 = (5q - 2b)(4q - 3b) \]