Answer :
To multiply the polynomials
$$
(4x^2 + 3x + 7)(8x - 5),
$$
we use the distributive property (also called the FOIL method for binomials) to multiply each term in the first polynomial by each term in the second polynomial.
1. Multiply the first term of the first polynomial by each term in the second polynomial:
$$
4x^2 \cdot 8x = 32x^3 \quad \text{and} \quad 4x^2 \cdot (-5) = -20x^2.
$$
2. Multiply the second term of the first polynomial by each term in the second polynomial:
$$
3x \cdot 8x = 24x^2 \quad \text{and} \quad 3x \cdot (-5) = -15x.
$$
3. Multiply the third term of the first polynomial by each term in the second polynomial:
$$
7 \cdot 8x = 56x \quad \text{and} \quad 7 \cdot (-5) = -35.
$$
4. Now, list all the products:
$$
32x^3,\quad -20x^2,\quad 24x^2,\quad -15x,\quad 56x,\quad -35.
$$
5. Combine like terms:
- The cubic term: $$32x^3.$$
- The quadratic terms: $$-20x^2 + 24x^2 = 4x^2.$$
- The linear terms: $$-15x + 56x = 41x.$$
- The constant term: $$-35.$$
Thus, the product is
$$
32x^3 + 4x^2 + 41x - 35.
$$
The correct answer is option D:
$$
32x^3+4x^2+41x-35.
$$
$$
(4x^2 + 3x + 7)(8x - 5),
$$
we use the distributive property (also called the FOIL method for binomials) to multiply each term in the first polynomial by each term in the second polynomial.
1. Multiply the first term of the first polynomial by each term in the second polynomial:
$$
4x^2 \cdot 8x = 32x^3 \quad \text{and} \quad 4x^2 \cdot (-5) = -20x^2.
$$
2. Multiply the second term of the first polynomial by each term in the second polynomial:
$$
3x \cdot 8x = 24x^2 \quad \text{and} \quad 3x \cdot (-5) = -15x.
$$
3. Multiply the third term of the first polynomial by each term in the second polynomial:
$$
7 \cdot 8x = 56x \quad \text{and} \quad 7 \cdot (-5) = -35.
$$
4. Now, list all the products:
$$
32x^3,\quad -20x^2,\quad 24x^2,\quad -15x,\quad 56x,\quad -35.
$$
5. Combine like terms:
- The cubic term: $$32x^3.$$
- The quadratic terms: $$-20x^2 + 24x^2 = 4x^2.$$
- The linear terms: $$-15x + 56x = 41x.$$
- The constant term: $$-35.$$
Thus, the product is
$$
32x^3 + 4x^2 + 41x - 35.
$$
The correct answer is option D:
$$
32x^3+4x^2+41x-35.
$$
