Answer :
To solve the problem of finding the product of [tex]\((3x^2 + 7)(6x^2 - 4x + 5)\)[/tex], we need to use the distributive property (also known as the FOIL method for binomials). This involves multiplying each term in the first polynomial by each term in the second polynomial and then combining like terms.
Here's a step-by-step solution:
1. Expand the product:
[tex]\[
(3x^2 + 7)(6x^2 - 4x + 5)
\][/tex]
2. Distribute [tex]\(3x^2\)[/tex] to each term in the second polynomial:
[tex]\[
3x^2 \cdot 6x^2 = 18x^4
\][/tex]
[tex]\[
3x^2 \cdot (-4x) = -12x^3
\][/tex]
[tex]\[
3x^2 \cdot 5 = 15x^2
\][/tex]
3. Distribute [tex]\(7\)[/tex] to each term in the second polynomial:
[tex]\[
7 \cdot 6x^2 = 42x^2
\][/tex]
[tex]\[
7 \cdot (-4x) = -28x
\][/tex]
[tex]\[
7 \cdot 5 = 35
\][/tex]
4. Combine all the terms from steps 2 and 3:
[tex]\[
18x^4 - 12x^3 + 15x^2 + 42x^2 - 28x + 35
\][/tex]
5. Combine like terms:
[tex]\[
18x^4 - 12x^3 + (15x^2 + 42x^2) - 28x + 35
\][/tex]
[tex]\[
18x^4 - 12x^3 + 57x^2 - 28x + 35
\][/tex]
Given the expanded and simplified polynomial, the correct answer is:
[tex]\[
\boxed{D. \ 18 x^4 - 12 x^3 + 57 x^2 - 28 x + 35}
\][/tex]
Here's a step-by-step solution:
1. Expand the product:
[tex]\[
(3x^2 + 7)(6x^2 - 4x + 5)
\][/tex]
2. Distribute [tex]\(3x^2\)[/tex] to each term in the second polynomial:
[tex]\[
3x^2 \cdot 6x^2 = 18x^4
\][/tex]
[tex]\[
3x^2 \cdot (-4x) = -12x^3
\][/tex]
[tex]\[
3x^2 \cdot 5 = 15x^2
\][/tex]
3. Distribute [tex]\(7\)[/tex] to each term in the second polynomial:
[tex]\[
7 \cdot 6x^2 = 42x^2
\][/tex]
[tex]\[
7 \cdot (-4x) = -28x
\][/tex]
[tex]\[
7 \cdot 5 = 35
\][/tex]
4. Combine all the terms from steps 2 and 3:
[tex]\[
18x^4 - 12x^3 + 15x^2 + 42x^2 - 28x + 35
\][/tex]
5. Combine like terms:
[tex]\[
18x^4 - 12x^3 + (15x^2 + 42x^2) - 28x + 35
\][/tex]
[tex]\[
18x^4 - 12x^3 + 57x^2 - 28x + 35
\][/tex]
Given the expanded and simplified polynomial, the correct answer is:
[tex]\[
\boxed{D. \ 18 x^4 - 12 x^3 + 57 x^2 - 28 x + 35}
\][/tex]
