Answer :
Let's multiply the polynomials [tex]\((4x^2 + 4x + 6)(7x + 5)\)[/tex] step-by-step.
First, we use the distributive property (often called the FOIL method for binomials) to expand the expression. We multiply each term in the first polynomial by each term in the second polynomial:
[tex]\[ (4x^2 + 4x + 6)(7x + 5) \][/tex]
### Step-by-Step Multiplication:
1. Multiply [tex]\(4x^2\)[/tex] by each term in [tex]\((7x + 5)\)[/tex]:
[tex]\[
4x^2 \cdot 7x = 28x^3
\][/tex]
[tex]\[
4x^2 \cdot 5 = 20x^2
\][/tex]
2. Multiply [tex]\(4x\)[/tex] by each term in [tex]\((7x + 5)\)[/tex]:
[tex]\[
4x \cdot 7x = 28x^2
\][/tex]
[tex]\[
4x \cdot 5 = 20x
\][/tex]
3. Multiply [tex]\(6\)[/tex] by each term in [tex]\((7x + 5)\)[/tex]:
[tex]\[
6 \cdot 7x = 42x
\][/tex]
[tex]\[
6 \cdot 5 = 30
\][/tex]
Now, we combine all these results:
[tex]\[
(4x^2 \cdot 7x) + (4x^2 \cdot 5) + (4x \cdot 7x) + (4x \cdot 5) + (6 \cdot 7x) + (6 \cdot 5)
\][/tex]
[tex]\[
= 28x^3 + 20x^2 + 28x^2 + 20x + 42x + 30
\][/tex]
Next, we combine like terms:
- The [tex]\(x^3\)[/tex] term: [tex]\( 28x^3 \)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\( 20x^2 + 28x^2 = 48x^2 \)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\( 20x + 42x = 62x \)[/tex]
- The constant term: [tex]\( 30 \)[/tex]
Putting it all together, we get:
[tex]\[
28x^3 + 48x^2 + 62x + 30
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{28x^3 + 48x^2 + 62x + 30}
\][/tex]
Hence, the correct option is:
A. [tex]\( 28x^3 + 48x^2 + 62x + 30 \)[/tex]
First, we use the distributive property (often called the FOIL method for binomials) to expand the expression. We multiply each term in the first polynomial by each term in the second polynomial:
[tex]\[ (4x^2 + 4x + 6)(7x + 5) \][/tex]
### Step-by-Step Multiplication:
1. Multiply [tex]\(4x^2\)[/tex] by each term in [tex]\((7x + 5)\)[/tex]:
[tex]\[
4x^2 \cdot 7x = 28x^3
\][/tex]
[tex]\[
4x^2 \cdot 5 = 20x^2
\][/tex]
2. Multiply [tex]\(4x\)[/tex] by each term in [tex]\((7x + 5)\)[/tex]:
[tex]\[
4x \cdot 7x = 28x^2
\][/tex]
[tex]\[
4x \cdot 5 = 20x
\][/tex]
3. Multiply [tex]\(6\)[/tex] by each term in [tex]\((7x + 5)\)[/tex]:
[tex]\[
6 \cdot 7x = 42x
\][/tex]
[tex]\[
6 \cdot 5 = 30
\][/tex]
Now, we combine all these results:
[tex]\[
(4x^2 \cdot 7x) + (4x^2 \cdot 5) + (4x \cdot 7x) + (4x \cdot 5) + (6 \cdot 7x) + (6 \cdot 5)
\][/tex]
[tex]\[
= 28x^3 + 20x^2 + 28x^2 + 20x + 42x + 30
\][/tex]
Next, we combine like terms:
- The [tex]\(x^3\)[/tex] term: [tex]\( 28x^3 \)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\( 20x^2 + 28x^2 = 48x^2 \)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\( 20x + 42x = 62x \)[/tex]
- The constant term: [tex]\( 30 \)[/tex]
Putting it all together, we get:
[tex]\[
28x^3 + 48x^2 + 62x + 30
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{28x^3 + 48x^2 + 62x + 30}
\][/tex]
Hence, the correct option is:
A. [tex]\( 28x^3 + 48x^2 + 62x + 30 \)[/tex]