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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]\(-40x^4 + 25x^3 + 45x^2\)[/tex]
B. [tex]\(-40x^3 + 25x^2 + 45x\)[/tex]
C. [tex]\(-40x^3 - 5x - 9\)[/tex]
D. [tex]\(-40x^2 + 25x + 45x\)[/tex]

Answer :

To find [tex]\((f \cdot g)(x)\)[/tex], we need to multiply the two functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] together.

Firstly, let’s understand the functions:

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

The expression [tex]\((f \cdot g)(x)\)[/tex] means [tex]\(f(x) \times g(x)\)[/tex]. So, let's compute:

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

Next, distribute [tex]\(-5x\)[/tex] to each term in [tex]\(g(x)\)[/tex]:

1. [tex]\(-5x \times 8x^2 = -40x^3\)[/tex]

2. [tex]\(-5x \times (-5x) = 25x^2\)[/tex]

3. [tex]\(-5x \times (-9) = 45x\)[/tex]

Now, putting it all together, the expanded form of the product is:

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

So the final answer is:
[tex]\(-40x^3 + 25x^2 + 45x\)[/tex]

Therefore, the correct choice from the options given is:
[tex]\(-40x^3 + 25x^2 + 45x\)[/tex]

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