https://file.aiquickdraw.com/mathbot/file/17266731031473g0w2p45.jpg

The expression to simplify is:

\[
\frac{45 - 5u^2}{u^2 - u - 6}
\]

Let's work through this step-by-step.

### Step 1: Factor both the numerator and the denominator
- **Numerator: \( 45 - 5u^2 \)**

This is a difference of squares. We can factor it as:

\[
45 - 5u^2 = 5(9 - u^2)
\]
Now, notice that \( 9 - u^2 \) is also a difference of squares:

\[
9 - u^2 = (3 - u)(3 + u)
\]

So the numerator becomes:

\[
5(3 - u)(3 + u)
\]

- **Denominator: \( u^2 - u - 6 \)**

We need to factor this quadratic expression. We look for two numbers that multiply to \(-6\) and add to \(-1\). These numbers are \(-3\) and \(2\), so we can factor it as:

\[
u^2 - u - 6 = (u - 3)(u + 2)
\]

### Step 2: Simplify the fraction

Now, substitute the factored forms of the numerator and denominator into the original expression:

\[
\frac{5(3 - u)(3 + u)}{(u - 3)(u + 2)}
\]

Notice that \( (3 - u) \) is the negative of \( (u - 3) \), so:

\[
(3 - u) = -(u - 3)
\]

Thus, the expression simplifies to:

\[
\frac{5(-(u - 3))(u + 3)}{(u - 3)(u + 2)} = -5 \cdot \frac{(u + 3)}{u + 2}
\]

### Final simplified form:

\[
\boxed{-5 \cdot \frac{u + 3}{u + 2}}
\]

Would you like further details on any of these steps? 

Here are some related questions for practice:
1. How do you factor a quadratic equation like \(u^2 - u - 6\)?
2. What is the difference of squares formula, and how is it applied here?
3. Can you simplify other rational expressions with higher powers?
4. How does recognizing patterns like \(a^2 - b^2\) help in simplifying algebraic expressions?
5. How do you handle similar simplifications when there are complex numbers involved?

**Tip**: When simplifying expressions, always check if common factors can be factored out before canceling terms to avoid mistakes.

Learn how to simplify the rational expression (45 - 5u^2) / (u^2 - u - 6) by factoring the numerator and denominator. This problem covers key concepts in algebra, including the difference of squares and factoring quadratics. Suitable for students in Grades 9-11.

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