Answer :
To multiply the polynomials
$$\left(7x^2+9x+7\right)(9x-4),$$
we expand the product by multiplying each term in the first polynomial by each term in the second polynomial.
1. Multiply each term in the first polynomial by $9x$:
\[
7x^2 \cdot 9x = 63x^3,
\]
\[
9x \cdot 9x = 81x^2,
\]
\[
7 \cdot 9x = 63x.
\]
2. Multiply each term in the first polynomial by $-4$:
\[
7x^2 \cdot (-4) = -28x^2,
\]
\[
9x \cdot (-4) = -36x,
\]
\[
7 \cdot (-4) = -28.
\]
3. Combine all the results:
\[
63x^3 + 81x^2 + 63x - 28x^2 - 36x - 28.
\]
4. Combine like terms:
- The $x^3$ term is simply:
\[
63x^3.
\]
- Add the $x^2$ terms:
\[
81x^2 - 28x^2 = 53x^2.
\]
- Add the $x$ terms:
\[
63x - 36x = 27x.
\]
- The constant term is:
\[
-28.
\]
Thus, the simplified expression is:
$$
63x^3 + 53x^2 + 27x - 28.
$$
This corresponds to option B.
$$\left(7x^2+9x+7\right)(9x-4),$$
we expand the product by multiplying each term in the first polynomial by each term in the second polynomial.
1. Multiply each term in the first polynomial by $9x$:
\[
7x^2 \cdot 9x = 63x^3,
\]
\[
9x \cdot 9x = 81x^2,
\]
\[
7 \cdot 9x = 63x.
\]
2. Multiply each term in the first polynomial by $-4$:
\[
7x^2 \cdot (-4) = -28x^2,
\]
\[
9x \cdot (-4) = -36x,
\]
\[
7 \cdot (-4) = -28.
\]
3. Combine all the results:
\[
63x^3 + 81x^2 + 63x - 28x^2 - 36x - 28.
\]
4. Combine like terms:
- The $x^3$ term is simply:
\[
63x^3.
\]
- Add the $x^2$ terms:
\[
81x^2 - 28x^2 = 53x^2.
\]
- Add the $x$ terms:
\[
63x - 36x = 27x.
\]
- The constant term is:
\[
-28.
\]
Thus, the simplified expression is:
$$
63x^3 + 53x^2 + 27x - 28.
$$
This corresponds to option B.
