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Answer :
To find the product of
[tex]$$
(6x^2 - 6x - 5)(7x^2 + 6x - 5),
$$[/tex]
we use the distributive property (also known as the FOIL method for quadratics). Follow these steps:
1. Multiply each term in the first polynomial by each term in the second polynomial:
- Multiply [tex]$6x^2$[/tex] by each term in the second polynomial:
[tex]$$
\begin{aligned}
6x^2 \cdot 7x^2 &= 42x^4, \\
6x^2 \cdot 6x &= 36x^3, \\
6x^2 \cdot (-5) &= -30x^2.
\end{aligned}
$$[/tex]
- Multiply [tex]$-6x$[/tex] by each term in the second polynomial:
[tex]$$
\begin{aligned}
(-6x) \cdot 7x^2 &= -42x^3, \\
(-6x) \cdot 6x &= -36x^2, \\
(-6x) \cdot (-5) &= 30x.
\end{aligned}
$$[/tex]
- Multiply [tex]$-5$[/tex] by each term in the second polynomial:
[tex]$$
\begin{aligned}
(-5) \cdot 7x^2 &= -35x^2, \\
(-5) \cdot 6x &= -30x, \\
(-5) \cdot (-5) &= 25.
\end{aligned}
$$[/tex]
2. Combine like terms:
- There is only one [tex]$x^4$[/tex] term:
[tex]$$
42x^4.
$$[/tex]
- Combine the [tex]$x^3$[/tex] terms:
[tex]$$
36x^3 - 42x^3 = -6x^3.
$$[/tex]
- Combine the [tex]$x^2$[/tex] terms:
[tex]$$
-30x^2 - 36x^2 - 35x^2 = -101x^2.
$$[/tex]
- Combine the [tex]$x$[/tex] terms:
[tex]$$
30x - 30x = 0.
$$[/tex]
- The constant term is:
[tex]$$
25.
$$[/tex]
3. Write the final result:
The product of the two polynomials is:
[tex]$$
42x^4 - 6x^3 - 101x^2 + 25.
$$[/tex]
Among the choices provided, this result corresponds to option B.
Thus, the correct answer is Option B.
[tex]$$
(6x^2 - 6x - 5)(7x^2 + 6x - 5),
$$[/tex]
we use the distributive property (also known as the FOIL method for quadratics). Follow these steps:
1. Multiply each term in the first polynomial by each term in the second polynomial:
- Multiply [tex]$6x^2$[/tex] by each term in the second polynomial:
[tex]$$
\begin{aligned}
6x^2 \cdot 7x^2 &= 42x^4, \\
6x^2 \cdot 6x &= 36x^3, \\
6x^2 \cdot (-5) &= -30x^2.
\end{aligned}
$$[/tex]
- Multiply [tex]$-6x$[/tex] by each term in the second polynomial:
[tex]$$
\begin{aligned}
(-6x) \cdot 7x^2 &= -42x^3, \\
(-6x) \cdot 6x &= -36x^2, \\
(-6x) \cdot (-5) &= 30x.
\end{aligned}
$$[/tex]
- Multiply [tex]$-5$[/tex] by each term in the second polynomial:
[tex]$$
\begin{aligned}
(-5) \cdot 7x^2 &= -35x^2, \\
(-5) \cdot 6x &= -30x, \\
(-5) \cdot (-5) &= 25.
\end{aligned}
$$[/tex]
2. Combine like terms:
- There is only one [tex]$x^4$[/tex] term:
[tex]$$
42x^4.
$$[/tex]
- Combine the [tex]$x^3$[/tex] terms:
[tex]$$
36x^3 - 42x^3 = -6x^3.
$$[/tex]
- Combine the [tex]$x^2$[/tex] terms:
[tex]$$
-30x^2 - 36x^2 - 35x^2 = -101x^2.
$$[/tex]
- Combine the [tex]$x$[/tex] terms:
[tex]$$
30x - 30x = 0.
$$[/tex]
- The constant term is:
[tex]$$
25.
$$[/tex]
3. Write the final result:
The product of the two polynomials is:
[tex]$$
42x^4 - 6x^3 - 101x^2 + 25.
$$[/tex]
Among the choices provided, this result corresponds to option B.
Thus, the correct answer is Option B.
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