https://file.aiquickdraw.com/mathbot/file/17254531820489gtmyjh5.jpg

To solve the problem, we are given two functions:

\[
f(x) = x^2 - 3x - 10, \quad g(x) = x - 5
\]

We need to find \((f \cdot g)(x)\), which represents the product of the two functions. This is calculated by multiplying \(f(x)\) and \(g(x)\):

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

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

\[
(f \cdot g)(x) = (x^2 - 3x - 10) \cdot (x - 5)
\]

Now, let's expand this expression:

\[
(x^2 - 3x - 10) \cdot (x - 5) = x^2(x) + x^2(-5) - 3x(x) - 3x(-5) - 10(x) - 10(-5)
\]

Simplify each term:

\[
= x^3 - 5x^2 - 3x^2 + 15x - 10x + 50
\]

Combine like terms:

\[
= x^3 - 8x^2 + 5x + 50
\]

So, the result for \((f \cdot g)(x)\) in its simplest form is:

\[
\boxed{x^3 - 8x^2 + 5x + 50}
\]

Would you like further details or have any questions?

Here are some related questions to explore:
1. How do you multiply polynomials in general?
2. What is the degree of a polynomial, and why is it important?
3. Can you factor the resulting polynomial \(x^3 - 8x^2 + 5x + 50\)?
4. How would you find the roots of \(f(x)\) or \(g(x)\) individually?
5. How can polynomial multiplication be visualized using an area model?

**Tip:** When multiplying polynomials, always distribute each term carefully to avoid missing any products!

Given that f(x) = x^2 - 3x - 10 and g(x) = x - 5, find (f ⋅ g)(x) and express the result as a polynomial in simplest form.

Learn how to multiply the polynomials f(x) = x^2 - 3x - 10 and g(x) = x - 5 and express the result in its simplest form. This problem involves algebraic concepts and the distributive property, suitable for students in Grades 9-11.

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