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We are given two functions:

\[
f(x) = 8x + 8
\]
and
\[
g(x) = 8x^2 + 2x.
\]

The problem asks for \((f \cdot g)(x)\), which is the product of the functions \(f(x)\) and \(g(x)\). This is computed by multiplying \(f(x)\) and \(g(x)\):

\[
(f \cdot g)(x) = f(x) \cdot g(x).
\]

Substitute the expressions for \(f(x)\) and \(g(x)\):

\[
(f \cdot g)(x) = (8x + 8)(8x^2 + 2x).
\]

Now, expand this product using the distributive property:

\[
= 8x(8x^2 + 2x) + 8(8x^2 + 2x).
\]

Let's compute each term:

\[
8x(8x^2 + 2x) = 64x^3 + 16x^2,
\]
\[
8(8x^2 + 2x) = 64x^2 + 16x.
\]

Now, add the terms together:

\[
(f \cdot g)(x) = 64x^3 + (16x^2 + 64x^2) + 16x.
\]

Combine like terms:

\[
(f \cdot g)(x) = 64x^3 + 80x^2 + 16x.
\]

So, the result is:

\[
(f \cdot g)(x) = 64x^3 + 80x^2 + 16x.
\]

Would you like more details or have any questions?

Here are five related questions to expand on this topic:
1. What is the difference between function addition and function multiplication?
2. How do you compute the derivative of the product of two functions?
3. What happens when you multiply functions involving higher-degree polynomials?
4. How would you compute \((g \cdot f)(x)\) instead of \((f \cdot g)(x)\)?
5. How does function composition differ from function multiplication?

**Tip**: When multiplying polynomials, always apply the distributive property carefully to avoid mistakes in expanding terms.

Let f(x) = 8x + 8 and g(x) = 8x^2 + 2x. Then, find (fg)(x).

Solve the polynomial function multiplication for f(x) = 8x + 8 and g(x) = 8x^2 + 2x, using distributive property to expand and combine like terms. This is suitable for students in grades 10-12.

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