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The expression you are asked to simplify is:

\[
\frac{9t^2 - 4}{\frac{3t^2 + 5t + 2}{t + 7}}
\]

### Step 1: Factor the expressions

1. **Factor the numerator** of the first fraction: \(9t^2 - 4\)
   - This is a difference of squares: 
   \[
   9t^2 - 4 = (3t - 2)(3t + 2)
   \]

2. **Factor the quadratic** in the numerator of the second fraction: \(3t^2 + 5t + 2\)
   - Factor the quadratic:
   \[
   3t^2 + 5t + 2 = (3t + 2)(t + 1)
   \]

Now, the expression becomes:

\[
\frac{(3t - 2)(3t + 2)}{\frac{(3t + 2)(t + 1)}{t + 7}}
\]

### Step 2: Simplify the division

When you divide by a fraction, you multiply by its reciprocal. So:

\[
\frac{(3t - 2)(3t + 2)}{\frac{(3t + 2)(t + 1)}{t + 7}} = (3t - 2)(3t + 2) \cdot \frac{t + 7}{(3t + 2)(t + 1)}
\]

### Step 3: Cancel common factors

The term \(3t + 2\) appears in both the numerator and denominator, so it cancels out:

\[
(3t - 2) \cdot \frac{t + 7}{t + 1}
\]

### Step 4: Final simplified expression

The final simplified expression is:

\[
\frac{(3t - 2)(t + 7)}{t + 1}
\]

Let me know if you would like more details or have any other questions!

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Here are 5 related questions for further practice:

1. How do you factor the expression \(x^2 - 9\)?
2. What is the general method for factoring quadratic expressions?
3. How can you simplify complex rational expressions?
4. Why is it important to check for common factors before multiplying fractions?
5. How does multiplying by the reciprocal work when dividing fractions?

### Tip: Always look for common factors to cancel before performing any multiplications when working with rational expressions!

Simplifica y escribe la contestación en su forma más simple: (9t^2 - 4) ÷ (3t^2 + 5t + 2)/(t + 7)

Learn how to simplify the rational expression (9t^2 - 4) ÷ (3t^2 + 5t + 2)/(t + 7) by factoring and canceling common terms. This step-by-step guide covers factoring quadratics, dividing fractions, and simplifying algebraic expressions. Suitable for high school students in Grades 9-12.

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