To simplify the expression \(\frac{x^2 - 12x + 20}{x - 2}\), let's first factor the quadratic in the numerator.
The quadratic \(x^2 - 12x + 20\) can be factored by looking for two numbers that multiply to \(20\) and add up to \(-12\). The numbers \(-10\) and \(-2\) fit this requirement. Thus, we can write:
\[ x^2 - 12x + 20 = (x - 10)(x - 2) \]
Now we can rewrite the original expression:
\[ \frac{x^2 - 12x + 20}{x - 2} = \frac{(x - 10)(x - 2)}{x - 2} \]
Assuming \(x \neq 2\) (since dividing by zero is not allowed), we can simplify:
\[ \frac{(x - 10)(x - 2)}{x - 2} = x - 10 \]
Therefore, the simplified expression is:
\[ x - 10 \quad \text{for } x \neq 2 \]