To find the binomial factors of the quadratic expression [tex]\(6s^2 + 40s - 64\)[/tex], we can follow the steps for factoring a quadratic expression of the form [tex]\(ax^2 + bx + c\)[/tex].
1. Identify the coefficients:
- [tex]\(a = 6\)[/tex]
- [tex]\(b = 40\)[/tex]
- [tex]\(c = -64\)[/tex]
2. Look for two numbers that multiply to [tex]\(a \cdot c\)[/tex] (which is [tex]\(6 \cdot -64 = -384\)[/tex]) and add to [tex]\(b\)[/tex] (which is [tex]\(40\)[/tex]).
The pair of numbers that meet these criteria is [tex]\(48\)[/tex] and [tex]\(-8\)[/tex], because:
[tex]\[
48 \cdot (-8) = -384 \quad \text{and} \quad 48 + (-8) = 40
\][/tex]
3. Rewrite the middle term using these numbers:
[tex]\[
6s^2 + 48s - 8s - 64
\][/tex]
4. Factor by grouping:
[tex]\[
6s(s + 8) - 8(s + 8)
\][/tex]
5. Factor out the common binomial factor:
[tex]\[
(6s - 8)(s + 8)
\][/tex]
6. Simplify the first binomial:
[tex]\[
(3s - 4)(s + 8)
\][/tex]
Therefore, the quadratic expression [tex]\(6s^2 + 40s - 64\)[/tex] factors into [tex]\((3s - 4)(s + 8)\)[/tex].
Hence, the binomial factors are [tex]\(3s - 4\)[/tex] and [tex]\(s + 8\)[/tex].
Among the provided choices, the best answer for one of the binomial factors of [tex]\(6s^2 + 40s - 64\)[/tex] is:
[tex]\[ \boxed{3s - 4} \][/tex]
