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To reduce the given rational expression:

\[
\frac{x^2 + 9x + 60}{x^2 + 8x + 16},
\]

### Step 1: Factorize the numerator (\(x^2 + 9x + 60\)):
We need two numbers that multiply to \(60\) and add to \(9\). These numbers are \(15\) and \(-4\).

Thus, the numerator factorizes as:
\[
x^2 + 9x + 60 = (x + 5)(x + 12).
\]

### Step 2: Factorize the denominator (\(x^2 + 8x + 16\)):
We need two numbers that multiply to \(16\) and add to \(8\). These numbers are \(4\) and \(4\).

Thus, the denominator factorizes as:
\[
x^2 + 8x + 16 = (x + 4)(x + 4) = (x + 4)^2.
\]

### Step 3: Write the factored form of the rational expression:
\[
\frac{x^2 + 9x + 60}{x^2 + 8x + 16} = \frac{(x + 5)(x + 12)}{(x + 4)^2}.
\]

### Step 4: Simplify (if possible):
There are no common factors in the numerator and denominator. Therefore, the simplified form remains:
\[
\frac{(x + 5)(x + 12)}{(x + 4)^2}.
\]

---

### Final Answer:
\[
\boxed{\frac{(x + 5)(x + 12)}{(x + 4)^2}}
\]

Let me know if you would like further explanation or assistance!

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#### 5 Related Questions:
1. How do you factorize a quadratic expression like \(x^2 + 9x + 60\)?
2. Why can’t we cancel terms like \(x + 4\) unless they are common factors?
3. How do you check if the factored form is correct?
4. What happens if there are common terms in the numerator and denominator?
5. How would the result change if the equation involved cubes instead of squares?

#### Tip:
Always verify your factored expressions by expanding them back to their original form to ensure accuracy.

Reduce the equation x^2 + 9x + 60 over x^2 + 8x + 16.

Learn how to reduce the rational expression (x^2 + 9x + 60) / (x^2 + 8x + 16) by factoring the numerator and denominator. This step-by-step solution explains factoring quadratics and simplifying rational expressions. Ideal for Grades 8-10 students studying algebra.

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