To determine the binomial factors of the quadratic polynomial [tex]\(6s^2 + 40s - 64\)[/tex], we need to factor it step-by-step. Here is the solution:
### Step 1: Write down the polynomial
[tex]\[ 6s^2 + 40s - 64 \][/tex]
### Step 2: Identify coefficients
From the quadratic polynomial [tex]\(ax^2 + bx + c\)[/tex]:
- [tex]\(a = 6\)[/tex]
- [tex]\(b = 40\)[/tex]
- [tex]\(c = -64\)[/tex]
### Step 3: Factor the polynomial using the quadratic formula
The quadratic formula is:
[tex]\[ s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Plugging in the values:
[tex]\[ s = \frac{-40 \pm \sqrt{40^2 - 4 \cdot 6 \cdot (-64)}}{2 \cdot 6} \][/tex]
[tex]\[ s = \frac{-40 \pm \sqrt{1600 + 1536}}{12} \][/tex]
[tex]\[ s = \frac{-40 \pm \sqrt{3136}}{12} \][/tex]
[tex]\[ s = \frac{-40 \pm 56}{12} \][/tex]
This gives us two solutions:
[tex]\[ s = \frac{-40 + 56}{12} = \frac{16}{12} = \frac{4}{3} \][/tex]
[tex]\[ s = \frac{-40 - 56}{12} = \frac{-96}{12} = -8 \][/tex]
### Step 4: Write the factors
The factors corresponding to these solutions are:
[tex]\[ s - \frac{4}{3} \][/tex]
[tex]\[ s + 8 \][/tex]
Since it's generally preferred to have integer coefficients, we can write the factored form as:
[tex]\[ 3( s - \frac{4}{3} ) = 3s - 4 \][/tex]
[tex]\[ s + 8 \][/tex]
Thus, our quadratic polynomial [tex]\(6s^2 + 40s - 64\)[/tex] can be factored as:
[tex]\[ (3s - 4)(2s + 16) \][/tex]
### Step 5: Simplification
We must verify that this is correct and comparable to our answer choices:
[tex]\[ 3s - 4 \text{ and } 2s + 16 \][/tex]
One of these factors fits within the answer choices:
- [tex]\(3s - 4\)[/tex] matches option D.
### Conclusion
The correct binomial factor from the given choices is [tex]\(3s - 4\)[/tex].
So, the answer is:
D. [tex]\(3s - 4\)[/tex]
