To simplify the expression
\[ \frac{y^2 - 3y + 2}{y^2 + 5y - 6}, \]
we can start by factoring both the numerator and the denominator.
Step 1: Factor the numerator \(y^2 - 3y + 2\)
We need to find two numbers that multiply to \(2\) (the constant term) and add to \(-3\) (the coefficient of \(y\)). The numbers \(-1\) and \(-2\) satisfy these conditions.
Thus, we can factor the numerator:
\[ y^2 - 3y + 2 = (y - 1)(y - 2). \]
Step 2: Factor the denominator \(y^2 + 5y - 6\)
Next, we look for two numbers that multiply to \(-6\) and add to \(5\). The numbers \(6\) and \(-1\) fit these requirements.
So, we factor the denominator:
\[ y^2 + 5y - 6 = (y + 6)(y - 1). \]
Step 3: Rewrite the expression
Now, we substitute the factored forms back into the expression:
\[ \frac{(y - 1)(y - 2)}{(y + 6)(y - 1)}. \]
Step 4: Simplify the expression
We notice that \((y - 1)\) appears in both the numerator and the denominator, and we can cancel it out (provided \(y \neq 1\)):
\[ \frac{(y - 2)}{(y + 6)} \quad \text{for } y \neq 1. \]
Final Result:
Thus, the simplified form of the expression is:
\[ \frac{y - 2}{y + 6} \quad \text{for } y \neq 1. \]