Step 1: Multiply numerators and denominators: (3x · 8)/(4 · 9x²) = (24x)/(36x²)

Step 2: Simplify by canceling common factors: (24x)/(36x²) = 24/(36x) = 2/(3x)

Step 3: Alternative - cancel before multiplying: (3x)/(4) · (8)/(9x²) = (3x · 8)/(4 · 9x²) Cancel 3 and 9: factor of 3 Cancel x and x²: one x Cancel 8 and 4: factor of 4 = (1 · 2)/(1 · 3x) = 2/(3x)

Step 4: Find restrictions: x ≠ 0

Answer: 2/(3x), where x ≠ 0

Multiply numerators and denominators: $\frac{(x + 2)(x - 3)}{(x - 3)(x + 5)}$

Cancel the common factor $(x - 3)$: $= \frac{x + 2}{x + 5}$

Answer: $\frac{x + 2}{x + 5}$

Step 1: Change division to multiplication: (x² - 4)/(x + 3) · (x² + 6x + 9)/(x + 2)

Step 2: Factor all expressions: x² - 4 = (x + 2)(x - 2) x² + 6x + 9 = (x + 3)²

Step 3: Rewrite with factors: [(x + 2)(x - 2)]/(x + 3) · [(x + 3)²]/(x + 2)

Step 4: Cancel common factors: Cancel (x + 2) and one (x + 3) = (x - 2) · (x + 3) = (x - 2)(x + 3)

Step 5: Expand (optional): = x² + 3x - 2x - 6 = x² + x - 6

Step 6: Find restrictions: x ≠ -3, -2 (from original denominators)

Answer: x² + x - 6 or (x - 2)(x + 3), where x ≠ -3, -2

Step 1: Multiply by the reciprocal $\frac{x^2 - 4}{x + 1} \cdot \frac{x^2 - 1}{x + 2}$

Step 2: Factor everything $\frac{(x + 2)(x - 2)}{x + 1} \cdot \frac{(x + 1)(x - 1)}{x + 2}$

Step 3: Cancel $(x + 2)$ and $(x + 1)$ $= \frac{(x - 2)(x - 1)}{1}$

Step 4: Multiply $= (x - 2)(x - 1) = x^2 - 3x + 2$

Answer: $x^2 - 3x + 2$

Step 1: Identify that denominators are the same: Both have denominator (x + 1)

Step 2: Add numerators: (2x + 3)/(x + 1)

Step 3: Check if numerator can be factored: 2x + 3 cannot be factored

Step 4: Find restrictions: x ≠ -1

Answer: (2x + 3)/(x + 1), where x ≠ -1

Step 1: Find the LCD: Denominators: 2x and 4x² LCD = 4x²

Step 2: Convert to equivalent fractions: (5)/(2x) = (5 · 2x)/(2x · 2x) = (10x)/(4x²) (3)/(4x²) already has the LCD

Step 3: Subtract numerators: (10x)/(4x²) - (3)/(4x²) = (10x - 3)/(4x²)

Step 4: Check if can be simplified: 10x - 3 cannot be factored with 4x²

Step 5: Find restrictions: x ≠ 0

Answer: (10x - 3)/(4x²), where x ≠ 0

Step 1: Find LCD $\text{LCD} = (x - 2)(x + 1)$

Step 2: Rewrite with LCD $\frac{3(x + 1)}{(x - 2)(x + 1)} + \frac{4(x - 2)}{(x - 2)(x + 1)}$

Step 3: Add numerators $= \frac{3(x + 1) + 4(x - 2)}{(x - 2)(x + 1)}$

Step 4: Expand and simplify $= \frac{3x + 3 + 4x - 8}{(x - 2)(x + 1)}$ $= \frac{7x - 5}{(x - 2)(x + 1)}$

Answer: $\frac{7x - 5}{(x - 2)(x + 1)}$

Step 1: Factor all denominators: x² - 1 = (x + 1)(x - 1) x² + 2x + 1 = (x + 1)² x + 1 = (x + 1)

Step 2: Find LCD: LCD = (x + 1)²(x - 1)

Step 3: Convert each fraction to LCD: x/[(x + 1)(x - 1)] = [x(x + 1)]/[(x + 1)²(x - 1)]

2/(x + 1)² = [2(x - 1)]/[(x + 1)²(x - 1)]

1/(x + 1) = [(x + 1)(x - 1)]/[(x + 1)²(x - 1)]

Step 4: Combine numerators: [x(x + 1) + 2(x - 1) - (x + 1)(x - 1)]/[(x + 1)²(x - 1)]

Step 5: Expand numerators: x² + x + 2x - 2 - (x² - 1) = x² + 3x - 2 - x² + 1 = 3x - 1

Step 6: Write final answer: (3x - 1)/[(x + 1)²(x - 1)]

Step 7: Find restrictions: x ≠ -1, 1

Answer: (3x - 1)/[(x + 1)²(x - 1)], where x ≠ -1, 1