To solve the problem of performing the operation and simplifying the product of the given fractions, let's go through a step-by-step solution.
We start with the following fractions:
[tex]\[
\frac{3x}{2x + 1} \cdot \frac{x + 1}{4x}
\][/tex]
Step 1: Multiply the numerators and the denominators.
Multiply the numerators:
[tex]\[
(3x) \cdot (x + 1) = 3x(x + 1) = 3x^2 + 3x
\][/tex]
Multiply the denominators:
[tex]\[
(2x + 1) \cdot 4x = 4x(2x + 1) = 8x^2 + 4x
\][/tex]
So, the resulting fraction after multiplication is:
[tex]\[
\frac{3x^2 + 3x}{8x^2 + 4x}
\][/tex]
Step 2: Simplify the fraction by factoring out the greatest common divisor (GCD).
Identify the greatest common divisor (GCD):
- For the numerator [tex]\(3x^2 + 3x\)[/tex], the GCD is [tex]\(3x\)[/tex].
- For the denominator [tex]\(8x^2 + 4x\)[/tex], the GCD is [tex]\(4x\)[/tex].
Factor out the GCD:
Numerator:
[tex]\[
3x^2 + 3x = 3x(x + 1)
\][/tex]
Denominator:
[tex]\[
8x^2 + 4x = 4x(2x + 1)
\][/tex]
So, the fraction is now:
[tex]\[
\frac{3x(x + 1)}{4x(2x + 1)}
\][/tex]
Step 3: Cancel any common factors in the numerator and the denominator.
In the fraction [tex]\(\frac{3x(x + 1)}{4x(2x + 1)}\)[/tex], we see that [tex]\(x\)[/tex] is a common factor in both the numerator and the denominator. Cancel the [tex]\(x\)[/tex]:
[tex]\[
\frac{3(x + 1)}{4(2x + 1)}
\][/tex]
Step 4: Write the simplified result.
The simplified fraction is:
[tex]\[
\frac{3(x + 1)}{4(2x + 1)}
\][/tex]
Therefore, the answer to the operation and simplification is:
[tex]\[
\boxed{\frac{3(x + 1)}{4(2x + 1)}}
\][/tex]
