To factor the trinomial [tex]\(8x^2 + 13x - 6\)[/tex], let's go through the process step by step:
1. Understand the Form: The trinomial is in the form [tex]\(ax^2 + bx + c\)[/tex], where [tex]\(a = 8\)[/tex], [tex]\(b = 13\)[/tex], and [tex]\(c = -6\)[/tex].
2. Find the Product and Sum: We need two numbers that multiply to [tex]\(a \cdot c = 8 \cdot (-6) = -48\)[/tex] and add up to [tex]\(b = 13\)[/tex].
3. List Factors of -48: Identify pairs of factors that multiply to -48:
- (-1, 48)
- (-2, 24)
- (-3, 16)
- (-4, 12)
- (-6, 8)
- (1, -48)
- (2, -24)
- (3, -16)
- (4, -12)
- (6, -8)
4. Find the Correct Pair: Look for the pair that adds up to [tex]\(b = 13\)[/tex]:
- The correct pair is (-3, 16) because [tex]\(-3 + 16 = 13\)[/tex].
5. Rewrite the Middle Term: Use the pair to split the middle term:
[tex]\[
8x^2 + 13x - 6 = 8x^2 - 3x + 16x - 6
\][/tex]
6. Group and Factor by Grouping:
[tex]\[
8x^2 - 3x + 16x - 6 = (8x^2 - 3x) + (16x - 6)
\][/tex]
Factor out the greatest common factor (GCF) from each group:
[tex]\[
= x(8x - 3) + 2(8x - 3)
\][/tex]
7. Factor Out the Common Binomial:
[tex]\[
= (x + 2)(8x - 3)
\][/tex]
So, the factorization of [tex]\(8x^2 + 13x - 6\)[/tex] is [tex]\((x + 2)(8x - 3)\)[/tex].
Thus, the correct answer is:
[tex]\[
(x + 2)(8x - 3)
\][/tex]
Final Answer:
[tex]\[
(x + 2)(8x - 3)
\][/tex]
