To solve \( (f \cdot g)(x) = (2x^2 - 4) \cdot (x^2 - 7x + 10) \), we'll perform the multiplication by distributing each term.

1. Start by expanding:
   \[
   (2x^2 - 4) \cdot (x^2 - 7x + 10)
   \]
   We can distribute each term of the first polynomial to the second polynomial.

2. First, distribute \( 2x^2 \) to each term of \( x^2 - 7x + 10 \):
   \[
   2x^2 \cdot x^2 = 2x^4
   \]
   \[
   2x^2 \cdot (-7x) = -14x^3
   \]
   \[
   2x^2 \cdot 10 = 20x^2
   \]

3. Next, distribute \( -4 \) to each term of \( x^2 - 7x + 10 \):
   \[
   -4 \cdot x^2 = -4x^2
   \]
   \[
   -4 \cdot (-7x) = 28x
   \]
   \[
   -4 \cdot 10 = -40
   \]

4. Now, combine all the terms together:
   \[
   2x^4 - 14x^3 + 20x^2 - 4x^2 + 28x - 40
   \]

5. Combine like terms:
   \[
   2x^4 - 14x^3 + (20x^2 - 4x^2) + 28x - 40
   \]
   \[
   2x^4 - 14x^3 + 16x^2 + 28x - 40
   \]

So, the simplified form of \( (f \cdot g)(x) \) is:
\[
2x^4 - 14x^3 + 16x^2 + 28x - 40
\]

Let me know if you'd like more details or if you have any questions!

Here are five related questions:
1. How do you factor polynomials like \( 2x^4 - 14x^3 + 16x^2 + 28x - 40 \)?
2. Can you solve for \( x \) when \( f(x) \cdot g(x) = 0 \)?
3. What is the degree of the polynomial \( f \cdot g(x) \)?
4. How does multiplying polynomials work in the context of functions?
5. How can you graph the function \( (f \cdot g)(x) \)?

**Tip:** When multiplying polynomials, always make sure to distribute each term carefully and then combine like terms at the end.

Learn how to multiply and simplify the polynomials (2x² - 4) and (x² - 7x + 10) step-by-step. This example uses the distributive property and combines like terms to simplify the expression. Suitable for students in Grades 9-11.

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