Certainly! Let's go through the steps to simplify the given expression and identify the non-permissible values.
### Given Expression
We need to simplify the expression:
[tex]\[
\frac{3 w^2 - 4 w - 4}{2 w - 4} \div \frac{3 w + 2}{w^2 + 5 w + 6}
\][/tex]
### Step-by-Step Simplification
1. Rewrite the Division as Multiplication by the Reciprocal:
To simplify the division, we can multiply the first fraction by the reciprocal of the second fraction:
[tex]\[
\frac{3 w^2 - 4 w - 4}{2 w - 4} \div \frac{3 w + 2}{w^2 + 5 w + 6} = \frac{3 w^2 - 4 w - 4}{2 w - 4} \times \frac{w^2 + 5 w + 6}{3 w + 2}
\][/tex]
2. Factorize where Possible:
For simpler manipulation, let's factorize the numerators and denominators where possible:
- The numerator [tex]\(3 w^2 - 4 w - 4\)[/tex] can be factorized:
[tex]\[
3 w^2 - 4 w - 4 = (3 w + 2)(w - 2)
\][/tex]
- The denominator [tex]\(2 w - 4\)[/tex] can be factorized:
[tex]\[
2 w - 4 = 2 (w - 2)
\][/tex]
- The denominator [tex]\(w^2 + 5 w + 6\)[/tex] can be factorized:
[tex]\[
w^2 + 5 w + 6 = (w + 3)(w + 2)
\][/tex]
So our expression is now:
[tex]\[
\frac{(3 w + 2)(w - 2)}{2 (w - 2)} \times \frac{(w + 3)(w + 2)}{3 w + 2}
\][/tex]
3. Cancel Out Common Factors:
Now we can cancel out the common factors in the numerators and denominators:
- The factor [tex]\( (3 w + 2) \)[/tex] appears in both a numerator and a denominator.
- The factor [tex]\( (w - 2) \)[/tex] also appears in both a numerator and a denominator.
After canceling these out, we are left with:
[tex]\[
\frac{(w + 3)(w + 2)}{2}
\][/tex]
4. Simplifying Further:
Multiplying the remaining terms, we get:
[tex]\[
\frac{(w + 3)(w + 2)}{2} = \frac{w^2 + 5 w + 6}{2}
\][/tex]
This is already simplified. Additionally, it can be written as:
[tex]\[
\frac{1}{2}(w^2 + 5w + 6)
\][/tex]
### Non-Permissible Values
Non-permissible values arise from the values of [tex]\( w \)[/tex] that make any denominator zero. We need to consider the denominators:
1. For [tex]\( \frac{3 w^2 - 4 w - 4}{2 w - 4} \)[/tex], the non-permissible value comes from [tex]\( 2 w - 4 \)[/tex]:
[tex]\[
2 w - 4 = 0 \implies w = 2
\][/tex]
2. For [tex]\( \frac{3 w + 2}{w^2 + 5 w + 6} \)[/tex], the non-permissible values come from [tex]\( w^2 + 5 w + 6 \)[/tex]:
[tex]\[
w^2 + 5 w + 6 = 0 \implies (w + 3)(w + 2) = 0
\implies w = -3 \quad \text{or} \quad w = -2
\][/tex]
Thus, the non-permissible values are [tex]\( w = 2, -3, -2 \)[/tex].
### Final Answer
The simplified form of the expression is:
[tex]\[
\frac{w^2 + 5w + 6}{2} \quad \text{or} \quad \frac{1}{2}(w^2 + 5w + 6)
\][/tex]
The non-permissible values are [tex]\( w = 2, w = -3, \text{ and } w = -2 \)[/tex].
