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The expression you need to multiply is:

\[
\frac{10x^2 + 13x - 3}{x^2 - 4x + 4} \cdot \frac{x^2 - 4}{5x^2 + 9x - 2}
\]

### Step-by-Step Solution:

1. **Factorize the expressions in the numerators and denominators** where possible.
   
   - The denominator of the first fraction is \( x^2 - 4x + 4 \), which is a perfect square trinomial and can be factored as:
     \[
     x^2 - 4x + 4 = (x - 2)^2
     \]

   - The numerator of the second fraction is \( x^2 - 4 \), which is a difference of squares and can be factored as:
     \[
     x^2 - 4 = (x - 2)(x + 2)
     \]

   - The quadratic expressions \( 10x^2 + 13x - 3 \) and \( 5x^2 + 9x - 2 \) require factoring by grouping or the quadratic formula:
     
     For \( 10x^2 + 13x - 3 \), factor it by grouping:
     \[
     10x^2 + 13x - 3 = (5x - 1)(2x + 3)
     \]
     For \( 5x^2 + 9x - 2 \), factor it as:
     \[
     5x^2 + 9x - 2 = (5x - 1)(x + 2)
     \]

2. **Rewrite the expression with these factors**:

   \[
   \frac{(5x - 1)(2x + 3)}{(x - 2)^2} \cdot \frac{(x - 2)(x + 2)}{(5x - 1)(x + 2)}
   \]

3. **Cancel out the common factors** from the numerator and denominator:

   - The factor \( (5x - 1) \) cancels out.
   - The factor \( (x - 2) \) cancels one of the terms from the first denominator \( (x - 2)^2 \).
   - The factor \( (x + 2) \) cancels out as well.

   After canceling, we are left with:
   \[
   \frac{2x + 3}{x - 2}
   \]

Thus, the result of the multiplication is:

\[
\boxed{\frac{2x + 3}{x - 2}}
\]

Would you like more details or have any other questions?

Here are 5 related questions for further exploration:

1. How do you factor general quadratic trinomials?
2. What are some methods to simplify complex algebraic fractions?
3. Can you always cancel terms in a fraction? What are the conditions?
4. How can the difference of squares be identified and factored?
5. What is the process for multiplying rational expressions in general?

**Tip:** Always check for common factors before multiplying fractions to simplify your work early.

Perform the following multiplication: (10x^2 + 13x - 3) / (x^2 - 4x + 4) * (x^2 - 4) / (5x^2 + 9x - 2)

Learn how to multiply rational expressions such as (10x^2 + 13x - 3) / (x^2 - 4x + 4) * (x^2 - 4) / (5x^2 + 9x - 2). This problem demonstrates factoring trinomials, recognizing the difference of squares, and simplifying complex algebraic fractions. Suitable for students in Grades 9-11.

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