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Answer :
To multiply
$$
(4x^2 + 4x + 6)(7x + 5),
$$
we use the distributive property (also known as the FOIL method for binomials) by multiplying every term in the first polynomial with every term in the second polynomial.
1. Multiply the term $4x^2$ by each term in $7x + 5$:
- $4x^2 \cdot 7x = 28x^3$
- $4x^2 \cdot 5 = 20x^2$
2. Multiply the term $4x$ by each term in $7x + 5$:
- $4x \cdot 7x = 28x^2$
- $4x \cdot 5 = 20x$
3. Multiply the term $6$ by each term in $7x + 5$:
- $6 \cdot 7x = 42x$
- $6 \cdot 5 = 30$
Now, combine like terms:
- The $x^3$ term is:
$$
28x^3.
$$
- The $x^2$ terms:
$$
20x^2 + 28x^2 = 48x^2.
$$
- The $x$ terms:
$$
20x + 42x = 62x.
$$
- The constant term:
$$
30.
$$
Thus, the product becomes:
$$
28x^3 + 48x^2 + 62x + 30.
$$
Comparing with the given multiple-choice options, the option that has the correct coefficients for $x^3$, $x^2$, and $x$ is the one that corresponds to choice C.
Therefore, the correct answer is C.
$$
(4x^2 + 4x + 6)(7x + 5),
$$
we use the distributive property (also known as the FOIL method for binomials) by multiplying every term in the first polynomial with every term in the second polynomial.
1. Multiply the term $4x^2$ by each term in $7x + 5$:
- $4x^2 \cdot 7x = 28x^3$
- $4x^2 \cdot 5 = 20x^2$
2. Multiply the term $4x$ by each term in $7x + 5$:
- $4x \cdot 7x = 28x^2$
- $4x \cdot 5 = 20x$
3. Multiply the term $6$ by each term in $7x + 5$:
- $6 \cdot 7x = 42x$
- $6 \cdot 5 = 30$
Now, combine like terms:
- The $x^3$ term is:
$$
28x^3.
$$
- The $x^2$ terms:
$$
20x^2 + 28x^2 = 48x^2.
$$
- The $x$ terms:
$$
20x + 42x = 62x.
$$
- The constant term:
$$
30.
$$
Thus, the product becomes:
$$
28x^3 + 48x^2 + 62x + 30.
$$
Comparing with the given multiple-choice options, the option that has the correct coefficients for $x^3$, $x^2$, and $x$ is the one that corresponds to choice C.
Therefore, the correct answer is C.
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