Let's first identify the excluded values. To do so, we need to find the values of [tex]\( y \)[/tex] that make any denominator in the expressions equal to zero. We have two denominators to consider:
1. [tex]\( 3y^2 - 12y \)[/tex]
2. [tex]\( y^2 + 2y - 8 \)[/tex]
Step 1: Find the excluded values for [tex]\( 3y^2 - 12y \)[/tex]:
Set the denominator equal to zero and solve for [tex]\( y \)[/tex]:
[tex]\[
3y^2 - 12y = 0
\][/tex]
Factor out the common term [tex]\( 3y \)[/tex]:
[tex]\[
3y(y - 4) = 0
\][/tex]
This gives us:
[tex]\[
3y = 0 \quad \text{or} \quad y - 4 = 0
\][/tex]
Solving these, we find:
[tex]\[
y = 0 \quad \text{or} \quad y = 4
\][/tex]
Step 2: Find the excluded values for [tex]\( y^2 + 2y - 8 \)[/tex]:
Set the denominator equal to zero and solve for [tex]\( y \)[/tex]:
[tex]\[
y^2 + 2y - 8 = 0
\][/tex]
Factor the quadratic:
[tex]\[
(y + 4)(y - 2) = 0
\][/tex]
This gives us:
[tex]\[
y = -4 \quad \text{or} \quad y = 2
\][/tex]
Combining all these, the excluded values are:
[tex]\[
y = 0, \, y = 4, \, y = -4, \, y = 2
\][/tex]
Step 3: Simplify the product:
Let's simplify each fraction individually first and then multiply them.
[tex]\[
\frac{6y^2 + 18y - 60}{3y^2 - 12y}
\][/tex]
Factor the numerator:
[tex]\[
6y^2 + 18y - 60 = 6(y^2 + 3y - 10) = 6(y + 5)(y - 2)
\][/tex]
Factor the denominator:
[tex]\[
3y^2 - 12y = 3y(y - 4)
\][/tex]
This simplifies to:
[tex]\[
\frac{6(y + 5)(y - 2)}{3y(y - 4)} = \frac{2(y + 5)(y - 2)}{y(y - 4)}
\][/tex]
Next fraction:
[tex]\[
\frac{y^2 - 16}{y^2 + 2y - 8}
\][/tex]
Factor the numerator:
[tex]\[
y^2 - 16 = (y - 4)(y + 4)
\][/tex]
Factor the denominator:
[tex]\[
y^2 + 2y - 8 = (y + 4)(y - 2)
\][/tex]
This simplifies to:
[tex]\[
\frac{(y - 4)(y + 4)}{(y + 4)(y - 2)} = \frac{y - 4}{y - 2}
\][/tex]
Multiplying the two simplified fractions:
[tex]\[
\frac{2(y + 5)(y - 2)}{y(y - 4)} \cdot \frac{y - 4}{y - 2} = \frac{2(y + 5) \cancel{(y - 2)} \cdot \cancel{(y - 4)}}{y \cdot \cancel{(y - 4)} \cdot \cancel{(y - 2)}}
\][/tex]
The cancels yield:
[tex]\[
\frac{2(y + 5)}{y}
\][/tex]
In simplest form:
[tex]\[
2 + \frac{10}{y}
\][/tex]
So, the product in its simplest form is:
[tex]\[
\boxed{2 + \frac{10}{y}}
\][/tex]
And the excluded values are [tex]\( y = 0, 2, 4, -4 \)[/tex].
