To simplify the expression \( \frac{2}{x(2x-3)} \cdot \frac{x(x+1)}{4} \), follow these steps:

1. Write the expression as a single fraction:
   \[
   \frac{2}{x(2x-3)} \cdot \frac{x(x+1)}{4} = \frac{2 \cdot x(x+1)}{x(2x-3) \cdot 4}
   \]

2. Simplify the numerator and the denominator separately:
   \[
   \text{Numerator: } 2 \cdot x(x+1) = 2x(x+1)
   \]
   \[
   \text{Denominator: } x(2x-3) \cdot 4 = 4x(2x-3)
   \]

3. Combine the simplified numerator and denominator:
   \[
   \frac{2x(x+1)}{4x(2x-3)}
   \]

4. Cancel the common factor \(x\) in the numerator and denominator:
   \[
   \frac{2 \cancel{x}(x+1)}{4 \cancel{x}(2x-3)} = \frac{2(x+1)}{4(2x-3)}
   \]

5. Simplify the remaining fraction by dividing both the numerator and the denominator by 2:
   \[
   \frac{2(x+1)}{4(2x-3)} = \frac{x+1}{2(2x-3)}
   \]

So, the simplified form of the expression is:
\[
\frac{x+1}{2(2x-3)}
\]

Would you like more details on any step? Here are five questions you can ask next:

1. How do I simplify complex fractions involving multiple terms?
2. Can you explain how to cancel common factors in fractions?
3. What are some common mistakes to avoid when simplifying algebraic expressions?
4. How do I simplify expressions with variables in both the numerator and denominator?
5. Can you provide examples of simplifying other algebraic fractions?

**Tip:** Always look for common factors to cancel in the numerator and denominator when simplifying fractions.

Learn how to simplify the algebraic expression 2/x(2x-3) x x(x+1)/4 with a detailed step-by-step guide. This tutorial covers key algebraic concepts such as fraction multiplication, factor cancellation, and simplification techniques. Suitable for students in Grades 7-9.

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