Sure, I'd be happy to guide you through solving the problem of multiplying and simplifying the given fractions. Let's look at the problem:
[tex]\[
\frac{1}{2x + 1} \times \frac{1}{2x - 1}
\][/tex]
### Step 1: Multiply the Fractions
When multiplying fractions, you multiply the numerators together and the denominators together. For our problem, the numerators are both 1, and the denominators are [tex]\(2x + 1\)[/tex] and [tex]\(2x - 1\)[/tex]:
[tex]\[
\frac{1 \cdot 1}{(2x + 1)(2x - 1)} = \frac{1}{(2x + 1)(2x - 1)}
\][/tex]
### Step 2: Simplify the Result
We need to simplify the expression [tex]\(\frac{1}{(2x + 1)(2x - 1)}\)[/tex].
Notice that the denominator is a product of two binomials. This can be recognized as a difference of squares:
[tex]\[
(a + b)(a - b) = a^2 - b^2
\][/tex]
In our case, [tex]\(a = 2x\)[/tex] and [tex]\(b = 1\)[/tex], hence:
[tex]\[
(2x + 1)(2x - 1) = (2x)^2 - 1^2
\][/tex]
Simplify this further:
[tex]\[
(2x)^2 - 1^2 = 4x^2 - 1
\][/tex]
### Step 3: Write the Final Simplified Expression
Now, substituting back into our fraction, we get:
[tex]\[
\frac{1}{4x^2 - 1}
\][/tex]
So, the simplified form of the original expression [tex]\(\frac{1}{2x + 1} \times \frac{1}{2x - 1}\)[/tex] is:
[tex]\[
\frac{1}{4x^2 - 1}
\][/tex]
### Summary
To summarize, the given multiplication of fractions:
[tex]\[
\frac{1}{2x + 1} \times \frac{1}{2x - 1}
\][/tex]
multiplies to form:
[tex]\[
\frac{1}{(2x + 1)(2x - 1)}
\][/tex]
which simplifies to:
[tex]\[
\frac{1}{4x^2 - 1}
\][/tex]
