We are given the following expression to simplify:

\[
\frac{3x^2 - 25x + 8}{2x^2 - 7x - 72}
\]

### Step 1: Factor the numerator
We have the quadratic expression in the numerator:

\[
3x^2 - 25x + 8
\]

We can try factoring this expression. We need two numbers that multiply to \(3 \times 8 = 24\) and add to \(-25\). These numbers are \(-24\) and \(-1\). So, we rewrite the middle term:

\[
3x^2 - 24x - x + 8
\]

Now, factor by grouping:

\[
3x(x - 8) - 1(x - 8) = (3x - 1)(x - 8)
\]

So, the numerator factors as:

\[
(3x - 1)(x - 8)
\]

### Step 2: Factor the denominator
Now, we factor the denominator \(2x^2 - 7x - 72\). We need two numbers that multiply to \(2 \times -72 = -144\) and add to \(-7\). These numbers are \(-16\) and \(9\). We rewrite the middle term:

\[
2x^2 - 16x + 9x - 72
\]

Now, factor by grouping:

\[
2x(x - 8) + 9(x - 8) = (2x + 9)(x - 8)
\]

So, the denominator factors as:

\[
(2x + 9)(x - 8)
\]

### Step 3: Simplify the expression
Now we have the factored form of the expression:

\[
\frac{(3x - 1)(x - 8)}{(2x + 9)(x - 8)}
\]

We can cancel out the common factor \((x - 8)\) from the numerator and denominator, as long as \(x \neq 8\).

The simplified expression is:

\[
\frac{3x - 1}{2x + 9}
\]

### Final Answer:

\[
\boxed{\frac{3x - 1}{2x + 9}}
\]

Let me know if you'd like more details or have questions!

Here are 5 related questions:
1. How do you solve rational equations with variables in the denominator?
2. What happens when there are no common factors to cancel in rational expressions?
3. How do you find the domain of a rational expression like this?
4. What are the steps to solve quadratic equations through factoring?
5. Can a rational expression like this one have any restrictions?

**Tip:** Always check for common factors in both the numerator and denominator when simplifying rational expressions!

Simplify the rational expression (3x^2 - 25x + 8) / (2x^2 - 7x - 72) by factoring both the numerator and denominator. Learn how to factor quadratics and simplify rational expressions in this step-by-step solution. Suitable for students in Grades 9-11.

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