To factor the trinomial [tex]\(12v^2 + 11v - 15\)[/tex], we can break it down step by step. Here's the process:
1. Identify coefficients and constants: In the trinomial [tex]\(12v^2 + 11v - 15\)[/tex]:
- The coefficient of [tex]\(v^2\)[/tex] is [tex]\(12\)[/tex].
- The coefficient of [tex]\(v\)[/tex] is [tex]\(11\)[/tex].
- The constant term is [tex]\(-15\)[/tex].
2. Split the middle term: We need to find two numbers that multiply to [tex]\(12 \times (-15)= -180\)[/tex] and add up to [tex]\(11\)[/tex]. The correct pair of numbers are [tex]\(15\)[/tex] and [tex]\(-4\)[/tex].
3. Rewrite and factor by grouping:
[tex]\[
12v^2 + 15v - 4v - 15
\][/tex]
Now, group the terms:
[tex]\[
(12v^2 + 15v) + (-4v - 15)
\][/tex]
4. Factor out the greatest common factor (GCF) from each group:
[tex]\[
3v(4v + 5) - 3(4v + 5)
\][/tex]
5. Factor out the common binomial factor:
[tex]\[
(4v + 5)(3v - 3)
\][/tex]
Therefore, the correct factorization of the trinomial [tex]\(12v^2 + 11v - 15\)[/tex] is:
[tex]\[
(4v + 5)(3v - 3)
\][/tex]
So the filled-in blanks should be:
[tex]\[
12v^2 + 11v - 15 = (4v + 5)(3v - 3)
\][/tex]
