https://file.aiquickdraw.com/mathbot/file/1729590552287e3ydr84d.jpg

The problem in the image is to find the quotient of two rational expressions. The given expression is:

\[
\frac{r + 4}{r^2 - 1} \div \frac{r^2 - 16}{r + 1}
\]

### Steps to solve:

1. **Rewrite the division as multiplication by the reciprocal:**

\[
\frac{r + 4}{r^2 - 1} \times \frac{r + 1}{r^2 - 16}
\]

2. **Factor the denominators** (both \( r^2 - 1 \) and \( r^2 - 16 \) are differences of squares):

- \( r^2 - 1 = (r - 1)(r + 1) \)
- \( r^2 - 16 = (r - 4)(r + 4) \)

So, the expression becomes:

\[
\frac{r + 4}{(r - 1)(r + 1)} \times \frac{r + 1}{(r - 4)(r + 4)}
\]

3. **Cancel out common factors:**
- The \( r + 4 \) cancels out.
- The \( r + 1 \) cancels out.

Now you're left with:

\[
\frac{1}{(r - 1)} \times \frac{1}{(r - 4)} = \frac{1}{(r - 1)(r - 4)}
\]

### Final Answer:
\[
\frac{1}{(r - 1)(r - 4)}
\]

Would you like more details or have any other questions?

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Here are 5 related questions to expand on this topic:

1. How do you factor quadratic expressions, such as \( r^2 - 1 \) and \( r^2 - 16 \)?
2. What are the general rules for dividing rational expressions?
3. How do you handle polynomials that can't be factored easily?
4. What happens if a term in the denominator is 0 in a rational expression?
5. How does the difference of squares factorization work?

**Tip:** When dividing rational expressions, always remember to multiply by the reciprocal and look for factors that can be canceled out.

Find the quotient of the rational expressions \( \frac{r+4}{r^2 - 1} \div \frac{r^2 - 16}{r+1} \).

Learn how to solve a division of rational expressions problem, involving factoring quadratic expressions and multiplying by the reciprocal. This problem is suitable for students in Grades 8-10 and covers key algebraic concepts.

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