https://file.aiquickdraw.com/mathbot/file/17254552170248p918dfa.jpg

Let's analyze the problem shown in the image.

The functions provided are:

1. \( h(x) = x^2 + 4x + 9 \)
2. \( g(x) = x^2 - 4x \)
3. \( f(x) = 3x + 9 \)

The question asks for an expression equivalent to \( \frac{h(x)}{g(x)} \).

Let's calculate \( \frac{h(x)}{g(x)} \):

Given:
\[ h(x) = x^2 + 4x + 9 \]
\[ g(x) = x^2 - 4x \]

The expression \( \frac{h(x)}{g(x)} \) becomes:
\[ \frac{x^2 + 4x + 9}{x^2 - 4x} \]

Now, let's simplify the expression if possible.

- The numerator \( x^2 + 4x + 9 \) does not factor nicely into simpler terms.
- The denominator \( x^2 - 4x \) can be factored as:
  \[
  x^2 - 4x = x(x - 4)
  \]
  
So, the expression becomes:
\[ \frac{x^2 + 4x + 9}{x(x - 4)} \]

None of the terms cancel out directly, so this is the simplest form.

Now, let's check the provided options for equivalence:

A. \(\frac{x + 4}{x}\)

B. \(\frac{-x + 4}{x}\)

C. \(\frac{x^2 + 4x + 9}{x(x - 4)}\) 

D. \(\frac{-x + 4}{x - 4}\)

The correct option that matches \( \frac{h(x)}{g(x)} \) is **C**.

Would you like further details or have any questions?

### Related Questions:
1. How would you factor the numerator \( x^2 + 4x + 9 \) if possible?
2. What would be the domain of the function \( \frac{h(x)}{g(x)} \)?
3. How would the expression change if \( g(x) = x^2 - 4 \) instead of \( x^2 - 4x \)?
4. What is the significance of factoring the denominator in expressions like these?
5. How can you verify the equivalence of two algebraic expressions?

### Tip:
When simplifying fractions involving polynomials, always check if the numerator and denominator can be factored to simplify the expression further.

Which expression is equal to \(\frac{h(x)}{g(x)}\)? Given functions: \(h(x) = x^2 + 4x + 9\), \(g(x) = x^2 - 4x\), and \(f(x) = 3x + 9\). Options: A. \(\frac{x + 4}{x}\), B. \(\frac{-x + 4}{x}\), C. \(\frac{x^2 + 4x + 9}{x(x - 4)}\), D. \(\frac{-x + 4}{x - 4}\).

Learn how to simplify the rational expression \(\frac{h(x)}{g(x)}\) with given functions \(h(x) = x^2 + 4x + 9\) and \(g(x) = x^2 - 4x\). This algebra problem covers concepts such as rational expressions and factoring, suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation