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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.

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 by factoring quadratics and canceling common terms. This example includes the expressions (10x^2 + 13x - 3)/(x^2 - 4x + 4) and (x^2 - 4)/(5x^2 + 9x - 2). Suitable for high school algebra students.

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