Sure, let's solve the given problem step by step.
1. Given Expressions:
[tex]\[
\frac{3x}{2x + 1} \quad \text{and} \quad \frac{x + 1}{4x}
\][/tex]
2. Multiplying the Fractions:
To multiply two fractions, you multiply the numerators together and the denominators together:
[tex]\[
\left( \frac{3x}{2x + 1} \right) \cdot \left( \frac{x + 1}{4x} \right) = \frac{3x \cdot (x + 1)}{(2x + 1) \cdot 4x}
\][/tex]
3. Simplifying the Expressions:
Multiply the numerators and the denominators separately:
[tex]\[
\frac{3x \cdot (x + 1)}{(2x + 1) \cdot 4x} = \frac{3x (x + 1)}{4x (2x + 1)}
\][/tex]
4. Simplify the Fraction:
- Notice that [tex]\(3x \cdot (x + 1)\)[/tex] and [tex]\(4x \cdot (2x + 1)\)[/tex] do not have common factors that can be simplified further.
- The x term in the numerator and denominator can cancel out one [tex]\( x \)[/tex] term:
[tex]\[
\frac{3(x + 1)}{4(2x + 1)}
\][/tex]
5. Final Simplified Expression:
Therefore, the simplified version of the fraction multiplication is:
[tex]\[
\frac{3(x + 1)}{4(2x + 1)}
\][/tex]
So, the result of the given operation is:
[tex]\[
\frac{3(x + 1)}{4(2x + 1)}
\][/tex]
