Sure! Let's work through the simplification step by step.
### Step 1: Factorize the denominator of the first fraction
Given expression:
[tex]\[
\frac{y-6}{y^2+4y-32} \cdot \frac{y-4}{y+2}
\][/tex]
First, we need to factorize the quadratic expression [tex]\(y^2 + 4y - 32\)[/tex]. Upon factoring, we get:
[tex]\[
y^2 + 4y - 32 = (y + 8)(y - 4)
\][/tex]
Thus, the fraction becomes:
[tex]\[
\frac{y-6}{(y+8)(y-4)} \cdot \frac{y-4}{y+2}
\][/tex]
### Step 2: Simplify the fractions
Next, we notice that there is a common term [tex]\((y - 4)\)[/tex] present in both the numerator and the denominator of the product of the two fractions. We can cancel out this common term:
[tex]\[
\frac{y-6}{(y+8) \cancel{(y-4)}} \cdot \frac{\cancel{(y-4)}}{y+2} = \frac{y-6}{y+8} \cdot \frac{1}{y+2}
\][/tex]
So, after cancellation, we get:
[tex]\[
\frac{y-6}{(y+8)(y+2)}
\][/tex]
### Step 3: Expand the simplified expression in the denominator
Now, we want to write the simplified result in a single fraction:
[tex]\[
\frac{y-6}{(y+2)(y+8)}
\][/tex]
This gives:
[tex]\[
\frac{y-6}{(y+2)(y+8)}
\][/tex]
So after simplification the answer is:
[tex]\[
\frac{y-6}{(y+2)(y+8)}
\][/tex]
Thus, your simplified expression is:
[tex]\[
\frac{y - 6}{(y + 2)(y + 8)}
\][/tex]
This is the final simplified form of the given expression.
