Answer :
To multiply the polynomials [tex]\((4x^2 + 4x + 6)\)[/tex] and [tex]\((7x + 5)\)[/tex], we'll use the distributive property. This means we'll multiply each term in the first polynomial by each term in the second polynomial and then combine like terms. Let's do it step-by-step:
1. Multiply each term in the first polynomial by each term in the second polynomial:
a. Multiply [tex]\(4x^2\)[/tex] by [tex]\(7x\)[/tex]:
[tex]\[
4x^2 \cdot 7x = 28x^3
\][/tex]
b. Multiply [tex]\(4x^2\)[/tex] by [tex]\(5\)[/tex]:
[tex]\[
4x^2 \cdot 5 = 20x^2
\][/tex]
c. Multiply [tex]\(4x\)[/tex] by [tex]\(7x\)[/tex]:
[tex]\[
4x \cdot 7x = 28x^2
\][/tex]
d. Multiply [tex]\(4x\)[/tex] by [tex]\(5\)[/tex]:
[tex]\[
4x \cdot 5 = 20x
\][/tex]
e. Multiply [tex]\(6\)[/tex] by [tex]\(7x\)[/tex]:
[tex]\[
6 \cdot 7x = 42x
\][/tex]
f. Multiply [tex]\(6\)[/tex] by [tex]\(5\)[/tex]:
[tex]\[
6 \cdot 5 = 30
\][/tex]
2. Combine all the terms:
[tex]\[
28x^3 + 20x^2 + 28x^2 + 20x + 42x + 30
\][/tex]
3. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(20x^2 + 28x^2 = 48x^2\)[/tex].
- Combine the [tex]\(x\)[/tex] terms: [tex]\(20x + 42x = 62x\)[/tex].
4. Write the final result:
[tex]\[
28x^3 + 48x^2 + 62x + 30
\][/tex]
Therefore, the result of multiplying the polynomials [tex]\((4x^2 + 4x + 6)\)[/tex] and [tex]\((7x + 5)\)[/tex] is [tex]\(\boxed{28x^3 + 48x^2 + 62x + 30}\)[/tex], which corresponds to option C.
1. Multiply each term in the first polynomial by each term in the second polynomial:
a. Multiply [tex]\(4x^2\)[/tex] by [tex]\(7x\)[/tex]:
[tex]\[
4x^2 \cdot 7x = 28x^3
\][/tex]
b. Multiply [tex]\(4x^2\)[/tex] by [tex]\(5\)[/tex]:
[tex]\[
4x^2 \cdot 5 = 20x^2
\][/tex]
c. Multiply [tex]\(4x\)[/tex] by [tex]\(7x\)[/tex]:
[tex]\[
4x \cdot 7x = 28x^2
\][/tex]
d. Multiply [tex]\(4x\)[/tex] by [tex]\(5\)[/tex]:
[tex]\[
4x \cdot 5 = 20x
\][/tex]
e. Multiply [tex]\(6\)[/tex] by [tex]\(7x\)[/tex]:
[tex]\[
6 \cdot 7x = 42x
\][/tex]
f. Multiply [tex]\(6\)[/tex] by [tex]\(5\)[/tex]:
[tex]\[
6 \cdot 5 = 30
\][/tex]
2. Combine all the terms:
[tex]\[
28x^3 + 20x^2 + 28x^2 + 20x + 42x + 30
\][/tex]
3. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(20x^2 + 28x^2 = 48x^2\)[/tex].
- Combine the [tex]\(x\)[/tex] terms: [tex]\(20x + 42x = 62x\)[/tex].
4. Write the final result:
[tex]\[
28x^3 + 48x^2 + 62x + 30
\][/tex]
Therefore, the result of multiplying the polynomials [tex]\((4x^2 + 4x + 6)\)[/tex] and [tex]\((7x + 5)\)[/tex] is [tex]\(\boxed{28x^3 + 48x^2 + 62x + 30}\)[/tex], which corresponds to option C.
