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Let's break down this problem step by step and factor the expressions to simplify the quotient.

### Step 1: Factor the numerator and denominator of both rational expressions.

1. **First Rational Expression:**

   \[
   \frac{x^2 - 81}{x^2 - 8x + 16}
   \]

   - The numerator \(x^2 - 81\) is a difference of squares, which factors as:
     \[
     x^2 - 81 = (x - 9)(x + 9)
     \]

   - The denominator \(x^2 - 8x + 16\) is a perfect square trinomial:
     \[
     x^2 - 8x + 16 = (x - 4)(x - 4) = (x - 4)^2
     \]

   So, the first rational expression becomes:
   \[
   \frac{(x - 9)(x + 9)}{(x - 4)^2}
   \]

2. **Second Rational Expression:**

   \[
   \frac{x^2 + x - 72}{x^2 + x - 20}
   \]

   - To factor the numerator \(x^2 + x - 72\), we need two numbers that multiply to \(-72\) and add to \(1\). These numbers are \(9\) and \(-8\):
     \[
     x^2 + x - 72 = (x + 9)(x - 8)
     \]

   - To factor the denominator \(x^2 + x - 20\), we need two numbers that multiply to \(-20\) and add to \(1\). These numbers are \(5\) and \(-4\):
     \[
     x^2 + x - 20 = (x + 5)(x - 4)
     \]

   So, the second rational expression becomes:
   \[
   \frac{(x + 9)(x - 8)}{(x + 5)(x - 4)}
   \]

### Step 2: Set up the division of the two rational expressions.

The problem asks for the quotient, which means dividing the first rational expression by the second one. Division of fractions is done by multiplying by the reciprocal of the second fraction:

\[
\frac{(x - 9)(x + 9)}{(x - 4)^2} \div \frac{(x + 9)(x - 8)}{(x + 5)(x - 4)}
\]

This becomes:

\[
\frac{(x - 9)(x + 9)}{(x - 4)^2} \times \frac{(x + 5)(x - 4)}{(x + 9)(x - 8)}
\]

### Step 3: Simplify the expression.

- Cancel out the common factors:
  - \( (x + 9) \) cancels out.
  - One \( (x - 4) \) from the numerator and denominator cancels out.

Thus, we are left with:

\[
\frac{(x - 9)(x + 5)}{(x - 4)(x - 8)}
\]

### Final Answer:

The simplified quotient is:

\[
\frac{(x - 9)(x + 5)}{(x - 4)(x - 8)}
\]

In the image, the missing values are \(+ 5\) for the numerator and \( - 4\) for the denominator.

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Would you like more details or have any questions?

### Related Questions:
1. How can you identify when to use factoring with rational expressions?
2. What is the difference between multiplying and dividing rational expressions?
3. How do you handle rational expressions with complex numbers in the denominator?
4. Why does the difference of squares factor in the form \((x - a)(x + a)\)?
5. How can we factor trinomials that don’t have obvious factors like the ones in this problem?

### Tip:
When factoring trinomials, always look for common factors first, and remember to check for difference of squares!

Use factoring to find the simplified quotient of these rational expressions: (x^2 - 81) / (x^2 - 8x + 16) ÷ (x^2 + x - 72) / (x^2 + x - 20).

Learn how to simplify rational expressions by factoring polynomials. This example involves factoring differences of squares and trinomials to find the simplified quotient of two rational expressions, suitable for students in Grades 9-12.

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