To multiply the polynomials \( (2x - 7)(3x^2 + 7x - 8) \), we will use the distributive property (also known as the FOIL method for binomials).
1. **Multiply \( 2x \) by each term in the second polynomial:**
- \( 2x \cdot 3x^2 = 6x^3 \)
- \( 2x \cdot 7x = 14x^2 \)
- \( 2x \cdot (-8) = -16x \)
2. **Multiply \(-7\) by each term in the second polynomial:**
- \(-7 \cdot 3x^2 = -21x^2\)
- \(-7 \cdot 7x = -49x\)
- \(-7 \cdot (-8) = 56\)
3. **Now, combine all these results:**
- \( 6x^3 + 14x^2 - 16x - 21x^2 - 49x + 56 \)
4. **Combine like terms:**
- For \(x^2\): \(14x^2 - 21x^2 = -7x^2\)
- For \(x\): \(-16x - 49x = -65x\)
So the resulting polynomial is:
\[
6x^3 - 7x^2 - 65x + 56
\]
Now, to fill in the coefficient responses:
- The value of the coefficient of the \(x^3\) term is **6**.
- The value of the coefficient of the \(x^2\) term is **-7**.
- The value of the coefficient of the \(x\) term is **-65**.
- The value of the coefficient of the constant is **56**.
So the final answers are:
- Coefficient of \(x^3\): **6**
- Coefficient of \(x^2\): **-7**
- Coefficient of \(x\): **-65**
- Coefficient of constant: **56**