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Given the functions:

[tex]
\[
\begin{array}{l}
f(x) = -5x \\
g(x) = 8x^2 - 5x - 9
\end{array}
\]
[/tex]

Find [tex](f \cdot g)(x)[/tex].

A. [tex]-40x^2 + 25x + 45x[/tex]

B. [tex]-40x^3 + 25x^2 + 45x[/tex]

C. [tex]-40x^4 \div 25x^3 + 45x^2[/tex]

D. [tex]-40x^3 - 5x - 9[/tex]

Answer :

We are given the functions

[tex]$$
f(x) = -5x \quad \text{and} \quad g(x) = 8x^2 - 5x - 9.
$$[/tex]

To find the product [tex]$(f \cdot g)(x)$[/tex], we multiply [tex]$f(x)$[/tex] by [tex]$g(x)$[/tex]:

[tex]$$
(f \cdot g)(x) = f(x) \cdot g(x) = (-5x)(8x^2 - 5x - 9).
$$[/tex]

Now, distribute [tex]$-5x$[/tex] to each term of [tex]$g(x)$[/tex]:

1. Multiply [tex]$-5x$[/tex] by [tex]$8x^2$[/tex]:
[tex]$$
-5x \cdot 8x^2 = -40x^3.
$$[/tex]

2. Multiply [tex]$-5x$[/tex] by [tex]$-5x$[/tex]:
[tex]$$
-5x \cdot (-5x) = 25x^2.
$$[/tex]

3. Multiply [tex]$-5x$[/tex] by [tex]$-9$[/tex]:
[tex]$$
-5x \cdot (-9) = 45x.
$$[/tex]

Now, combine the results:

[tex]$$
(f \cdot g)(x) = -40x^3 + 25x^2 + 45x.
$$[/tex]

Thus, the final answer is:

[tex]$$
\boxed{-40x^3 + 25x^2 + 45x}.
$$[/tex]

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