Answer :
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].
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].