The expression in the image asks to perform the multiplication:

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

Let's go through the steps for solving this multiplication:

Step 1: Factor all polynomials, if possible.

1. Numerator of the first fraction: $10x^2 + 13x - 3$:

   • This is a quadratic expression. We look for factors of $10 \times (-3) = -30$ that add to $13$. The factors are $15$ and $-2$, so we rewrite and factor by grouping: $10x^2 + 15x - 2x - 3 = 5x(2x + 3) - 1(2x + 3) = (5x - 1)(2x + 3)$

2. Denominator of the first fraction: $x^2 - 4x + 4$:

   • This is a perfect square trinomial: $x^2 - 4x + 4 = (x - 2)^2$

3. Numerator of the second fraction: $x^2 - 4$:

   • This is a difference of squares: $x^2 - 4 = (x - 2)(x + 2)$

4. Denominator of the second fraction: $5x^2 + 9x - 2$:

   • We factor this quadratic expression. We look for factors of $5 \times (-2) = -10$ that add to $9$, which are $10$ and $-1$: $5x^2 + 10x - x - 2 = 5x(x + 2) - 1(x + 2) = (5x - 1)(x + 2)$

Step 2: Multiply the factored forms.

Now we rewrite the entire expression in factored form: $\frac{(5x - 1)(2x + 3)}{(x - 2)^2} \cdot \frac{(x - 2)(x + 2)}{(5x - 1)(x + 2)}$

Step 3: Cancel common factors.

• The factor $(5x - 1)$ appears in both the numerator and denominator, so it cancels out.
• The factor $(x - 2)$ appears in both the numerator and denominator, so one of them cancels out with one of the $(x - 2)$ factors in the denominator.
• The factor $(x + 2)$ appears in both the numerator and denominator, so it cancels out.

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

Final Answer:

The result of the multiplication is: $\frac{2x + 3}{x - 2}$

Let me know if you need further clarifications or if you'd like additional details!

Here are 5 related questions to further your understanding:

1. How do you factor a trinomial like $ax^2 + bx + c$?
2. What is the difference of squares and how is it factored?
3. Why do common factors cancel out in a rational expression?
4. Can all quadratics be factored? If not, what methods are used to solve them?
5. How can you check your work after multiplying rational expressions?

Tip: Always look for common factors in both the numerator and denominator before multiplying expressions, as it simplifies the process significantly.