Find the (gh)(x). 

Given:g(x) =(2x²-x+3) 
            h(x) =(4x²+5x)

To find \((gh)(x)\), we need to multiply the functions \(g(x)\) and \(h(x)\):

Given:
\[
g(x) = 2x^2 - x + 3
\]
\[
h(x) = 4x^2 + 5x
\]

The product \((gh)(x)\) is obtained by multiplying \(g(x)\) and \(h(x)\):

\[
(gh)(x) = g(x) \cdot h(x) = (2x^2 - x + 3)(4x^2 + 5x)
\]

Now, let's multiply the expressions step-by-step:

\[
(gh)(x) = (2x^2)(4x^2) + (2x^2)(5x) + (-x)(4x^2) + (-x)(5x) + (3)(4x^2) + (3)(5x)
\]

Simplify each term:

\[
(gh)(x) = 8x^4 + 10x^3 - 4x^3 - 5x^2 + 12x^2 + 15x
\]

Combine like terms:

\[
(gh)(x) = 8x^4 + (10x^3 - 4x^3) + (-5x^2 + 12x^2) + 15x
\]
\[
(gh)(x) = 8x^4 + 6x^3 + 7x^2 + 15x
\]

Therefore, the product of \(g(x)\) and \(h(x)\) is:

\[
(gh)(x) = 8x^4 + 6x^3 + 7x^2 + 15x
\]

### Would you like more details, or have any questions?

#### Related Questions:
1. What is the sum of two polynomial functions?
2. How do you find the quotient of two polynomial functions?
3. How do you perform polynomial long division?
4. What are the roots of the function \(g(x)\)?
5. How can we find the derivative of the function \((gh)(x)\)?

#### Tip:
When multiplying polynomials, ensure you multiply each term of one polynomial by every term of the other polynomial, then combine like terms.

Find the (gh)(x). Given: g(x) = 2x² - x + 3, h(x) = 4x² + 5x

Learn how to find the product of two polynomial functions, g(x) = 2x² - x + 3 and h(x) = 4x² + 5x. Follow the step-by-step solution using polynomial multiplication and the distributive property. Ideal for high school students, Grades 9-12.

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