Sure, let's go through each step to factor the polynomial [tex]\(16x^2 + 8x - 15\)[/tex] completely.
1. Write down the polynomial:
[tex]\[
16x^2 + 8x - 15
\][/tex]
2. Identify terms:
- [tex]\(a = 16\)[/tex], the coefficient of [tex]\(x^2\)[/tex]
- [tex]\(b = 8\)[/tex], the coefficient of [tex]\(x\)[/tex]
- [tex]\(c = -15\)[/tex], the constant term
3. Find two numbers that multiply to [tex]\(a \cdot c\)[/tex] (product) and add up to [tex]\(b\)[/tex] (sum):
- Here, [tex]\(a \cdot c = 16 \cdot (-15) = -240\)[/tex]
- We need two numbers that multiply to [tex]\(-240\)[/tex] and add to [tex]\(8\)[/tex].
After finding, we determine these numbers to be [tex]\(20\)[/tex] and [tex]\(-12\)[/tex] since:
[tex]\[
20 \times (-12) = -240
\][/tex]
[tex]\[
20 + (-12) = 8
\][/tex]
4. Rewrite the middle term using these numbers:
[tex]\[
16x^2 + 20x - 12x - 15
\][/tex]
5. Factor by grouping:
- Group the terms in pairs:
[tex]\[
(16x^2 + 20x) + (-12x - 15)
\][/tex]
- Factor out the greatest common factor (GCF) from each pair:
[tex]\[
4x(4x + 5) - 3(4x + 5)
\][/tex]
6. Factor out the common binomial factor:
[tex]\[
(4x + 5)(4x - 3)
\][/tex]
Thus, the factorization of [tex]\(16x^2 + 8x - 15\)[/tex] is:
[tex]\[
(4x + 5)(4x - 3)
\][/tex]
Place this answer in the provided location on the grid:
[tex]\[
(4x + 5)(4x - 3)
\][/tex]
