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To simplify the expression \(\frac{x^2 - 16x + 63}{x^2 - 14x + 45}\), we should start by factoring both the numerator and the denominator.

### Step 1: Factor the Numerator
The numerator is \(x^2 - 16x + 63\). We need to find two numbers that multiply to \(63\) and add up to \(-16\).
- The factors of \(63\) that sum up to \(-16\) are \(-9\) and \(-7\).
- So, \(x^2 - 16x + 63 = (x - 9)(x - 7)\).

### Step 2: Factor the Denominator
The denominator is \(x^2 - 14x + 45\). We need to find two numbers that multiply to \(45\) and add up to \(-14\).
- The factors of \(45\) that sum up to \(-14\) are \(-9\) and \(-5\).
- So, \(x^2 - 14x + 45 = (x - 9)(x - 5)\).

### Step 3: Simplify the Expression
Now we substitute these factored forms back into the expression:
\[
\frac{x^2 - 16x + 63}{x^2 - 14x + 45} = \frac{(x - 9)(x - 7)}{(x - 9)(x - 5)}
\]

We can cancel the common factor \((x - 9)\) from the numerator and the denominator, provided \(x \neq 9\) to avoid division by zero.

This leaves us with:
\[
\frac{x - 7}{x - 5}
\]

### Final Answer
\[
\frac{x - 7}{x - 5}
\]

Would you like further details on any part of this process, or have any questions?

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Here are some related questions for further exploration:
1. How do you factor trinomials like \(x^2 - 16x + 63\)?
2. What happens if there is no common factor to cancel in a rational expression?
3. How can you determine if a rational expression has any restrictions on \(x\)?
4. How would this simplification change if we had different constants in the trinomial terms?
5. How can we verify the answer by substituting values of \(x\) into the original and simplified expressions?

**Tip:** Always check for any common factors in the numerator and denominator that can be canceled to simplify expressions.

Simplify the expression \(\frac{x^2 - 16x + 63}{x^2 - 14x + 45}\).

Learn how to simplify the rational expression \(\frac{x^2 - 16x + 63}{x^2 - 14x + 45}\) by factoring both the numerator and denominator. This problem covers essential algebraic concepts in factoring and rational expressions, suitable for Grades 9-10.

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