Answer :
Let's find the product of the two polynomials [tex]\((3x^2 + 7)(6x^2 - 4x + 5)\)[/tex].
1. Distribute the terms from the first polynomial:
- First, 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]
Now we have:
[tex]\[
18x^4 - 12x^3 + 15x^2
\][/tex]
2. Distribute the second term from the first polynomial, 7, 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]
Now we have:
[tex]\[
42x^2 - 28x + 35
\][/tex]
3. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(15x^2 + 42x^2 = 57x^2\)[/tex]
4. Write the final expression:
The full polynomial expression is:
[tex]\[
18x^4 - 12x^3 + 57x^2 - 28x + 35
\][/tex]
After performing these steps, we can see that the correct answer is [tex]\(\boxed{18x^4 - 12x^3 + 57x^2 - 28x + 35}\)[/tex], which corresponds to option D.
1. Distribute the terms from the first polynomial:
- First, 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]
Now we have:
[tex]\[
18x^4 - 12x^3 + 15x^2
\][/tex]
2. Distribute the second term from the first polynomial, 7, 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]
Now we have:
[tex]\[
42x^2 - 28x + 35
\][/tex]
3. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(15x^2 + 42x^2 = 57x^2\)[/tex]
4. Write the final expression:
The full polynomial expression is:
[tex]\[
18x^4 - 12x^3 + 57x^2 - 28x + 35
\][/tex]
After performing these steps, we can see that the correct answer is [tex]\(\boxed{18x^4 - 12x^3 + 57x^2 - 28x + 35}\)[/tex], which corresponds to option D.
