Question: top-left: The text “The rational expression” is on the left.
center: The fraction $\frac{x^2 - 9x + 20}{x^2 - 3x - 10}$ appears in the middle.
right: The text “simplifies to _________.” appears on the right.
1. **State the problem:** Simplify the rational expression $$\frac{x^2 - 9x + 20}{x^2 - 3x - 10}$$.
2. **Recall the formula and rules:** To simplify a rational expression, factor both numerator and denominator completely and then cancel any common factors.
3. **Factor the numerator:**
$$x^2 - 9x + 20 = (x - 5)(x - 4)$$
4. **Factor the denominator:**
$$x^2 - 3x - 10 = (x - 5)(x + 2)$$
5. **Rewrite the expression with factors:**
$$\frac{(x - 5)(x - 4)}{(x - 5)(x + 2)}$$
6. **Cancel the common factor $(x - 5)$:**
$$\frac{\cancel{(x - 5)}(x - 4)}{\cancel{(x - 5)}(x + 2)} = \frac{x - 4}{x + 2}$$
7. **Final simplified form:**
$$\frac{x - 4}{x + 2}$$
**Note:** The expression is undefined at $x = 5$ and $x = -2$ because these values make the original denominator zero.