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.