Sure, let's simplify the given expression step by step.
### Expression to Simplify
We need to simplify:
[tex]\[
\frac{16 x^2 - 4}{4 x^2 + 2 x - 2} \div \frac{2 x + 1}{x + 1}
\][/tex]
### Step 1: Rewriting the Division as Multiplication
Remember that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we can rewrite the expression as:
[tex]\[
\frac{16 x^2 - 4}{4 x^2 + 2 x - 2} \cdot \frac{x + 1}{2 x + 1}
\][/tex]
### Step 2: Factor the Numerator and Denominator
Let's factor each part of the expressions.
1. Factor the numerator [tex]\(16 x^2 - 4\)[/tex]:
[tex]\[
16 x^2 - 4 = 4(4 x^2 - 1) = 4(2 x - 1)(2 x + 1)
\][/tex]
2. Factor the denominator [tex]\(4 x^2 + 2 x - 2\)[/tex]:
This is a bit more complex, but we can use factoring by grouping or the quadratic formula. Factoring, we get:
[tex]\[
4 x^2 + 2 x - 2 = (2 x + 2)(2 x - 1)
\][/tex]
### Step 3: Rewrite the Expression with Factored Forms
Now we can rewrite the original expression with these factored forms:
[tex]\[
\frac{4 (2 x - 1)(2 x + 1)}{(2 x + 2)(2 x - 1)} \cdot \frac{x + 1}{2 x + 1}
\][/tex]
### Step 4: Cancel Common Factors
Observe that [tex]\((2x - 1)\)[/tex] and [tex]\((2x + 1)\)[/tex] appear in both the numerator and the denominator. We can cancel these common factors:
[tex]\[
\frac{4 \cancel{(2 x - 1)} \cancel{(2 x + 1)}}{\cancel{(2 x + 1)} (2 x + 2)} \cdot \frac{x + 1}{\cancel{2 x + 1}}
\][/tex]
We are left with:
[tex]\[
\frac{4}{2 x + 2} \cdot (x + 1)
\][/tex]
### Step 5: Further Simplification
Notice that the denominator [tex]\(2 x + 2\)[/tex] can be factored as [tex]\(2(x + 1)\)[/tex]:
[tex]\[
\frac{4}{2(x + 1)} \cdot (x + 1)
\][/tex]
Now, we can cancel [tex]\((x + 1)\)[/tex] in the numerator and the denominator:
[tex]\[
\frac{4}{2} = 2
\][/tex]
Thus, the simplified form of the expression is:
[tex]\[
\boxed{2}
\][/tex]
