https://file.aiquickdraw.com/mathbot/file/172538433549759qy53wd.jpg

The problem in the image asks to divide the given rational expressions and express the quotient in its simplest form:

\[
\frac{\frac{25x^2 - 20x + 9}{5x^3 + 27x - 18}}{\frac{25x^2 - 9}{5x^3 + 25x + 12}}
\]

### Step 1: Simplify Each Expression

1. **Simplify the Numerators**:
   - \( 25x^2 - 20x + 9 \): This quadratic expression doesn't factor neatly into integers, so we'll leave it as it is.
   - \( 25x^2 - 9 \): This can be factored as a difference of squares:
     \[
     25x^2 - 9 = (5x - 3)(5x + 3)
     \]

2. **Simplify the Denominators**:
   - \( 5x^3 + 27x - 18 \): This cubic expression is complex, and there are no simple factors, so we'll leave it in its original form.
   - \( 5x^3 + 25x + 12 \): Again, this cubic expression doesn't factor easily, so it remains unchanged.

### Step 2: Write the Expression as a Multiplication Problem

The division of two fractions can be converted into multiplication by flipping the second fraction:

\[
\frac{25x^2 - 20x + 9}{5x^3 + 27x - 18} \times \frac{5x^3 + 25x + 12}{25x^2 - 9}
\]

### Step 3: Substitute the Simplified Terms

Substitute the factored forms:

\[
\frac{25x^2 - 20x + 9}{5x^3 + 27x - 18} \times \frac{5x^3 + 25x + 12}{(5x - 3)(5x + 3)}
\]

### Step 4: Look for Common Factors

Since there are no common factors in the numerator and denominator that can be canceled out, this expression is already in its simplest form.

Thus, the final simplified expression is:

\[
\frac{(25x^2 - 20x + 9)(5x^3 + 25x + 12)}{(5x^3 + 27x - 18)(5x - 3)(5x + 3)}
\]

This is the quotient expressed in its simplest form.

Would you like further details or have any questions?

Here are 5 related questions for further practice:

1. How do you factor a quadratic expression such as \(25x^2 - 20x + 9\)?
2. What is the process for factoring a cubic polynomial like \(5x^3 + 27x - 18\)?
3. How can the difference of squares be applied to more complex expressions?
4. What are some techniques for simplifying complex rational expressions?
5. How do you handle division of rational expressions involving polynomials?

**Tip:** Always check if any polynomial can be factored before attempting to simplify or divide rational expressions. This can significantly reduce the complexity of the problem.

Divide the rational expressions and express the quotient in simplest form:

\(
\frac{25x^2 - 20x + 9}{5x^3 + 27x - 18} \div \frac{25x^2 - 9}{5x^3 + 25x + 12}
\)

Learn how to divide and simplify rational expressions by factoring polynomials and applying the difference of squares formula. This detailed solution is ideal for high school students and covers key algebraic concepts.

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