Certainly! Let's simplify the given rational expression step-by-step:
Given:
[tex]\[ \frac{y^2 - y - 12}{y^2 - 2y - 15} \][/tex]
Step 1: Factor the numerator and the denominator.
First, factor the numerator [tex]\( y^2 - y - 12 \)[/tex].
Look for two numbers that multiply to [tex]\(-12\)[/tex] (the constant term) and add up to [tex]\(-1\)[/tex] (the coefficient of the middle term).
Those two numbers are [tex]\( -4 \)[/tex] and [tex]\( 3 \)[/tex].
So the numerator factors as:
[tex]\[ y^2 - y - 12 = (y - 4)(y + 3) \][/tex]
Next, factor the denominator [tex]\( y^2 - 2 y - 15 \)[/tex].
Look for two numbers that multiply to [tex]\( -15 \)[/tex] (the constant term) and add up to [tex]\( -2 \)[/tex] (the coefficient of the middle term).
Those two numbers are [tex]\( -5 \)[/tex] and [tex]\( 3 \)[/tex].
So the denominator factors as:
[tex]\[ y^2 - 2 y - 15 = (y - 5)(y + 3) \][/tex]
Step 2: Write the fraction with the factored forms.
Now, substitute the factored forms into the original expression:
[tex]\[ \frac{(y - 4)(y + 3)}{(y - 5)(y + 3)} \][/tex]
Step 3: Simplify the fraction by canceling common factors.
Notice that [tex]\( (y + 3) \)[/tex] is a common factor in both the numerator and the denominator, so we can cancel them out:
[tex]\[ \frac{(y - 4) \cancel{(y + 3)}}{(y - 5) \cancel{(y + 3)}} = \frac{y - 4}{y - 5} \][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[ \frac{y - 4}{y - 5} \][/tex]
So, the correct answer is:
[tex]\[ \boxed{\frac{y - 4}{y - 5}} \][/tex]
