Steps:

1. Factor everything
2. Multiply numerators and denominators
3. Cancel common factors
4. Simplify

Dividing Rational Expressions

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$

Multiply by the reciprocal!

Adding/Subtracting (Same Denominator)

$\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}$

Combine numerators, keep denominator.

Adding/Subtracting (Different Denominators)

1. Find the LCD (Least Common Denominator)
2. Rewrite each fraction with the LCD
3. Add or subtract numerators
4. Simplify

Example: $\frac{2}{x} + \frac{3}{x + 1}$

LCD = $x(x + 1)$

$\frac{2(x + 1)}{x(x + 1)} + \frac{3x}{x(x + 1)} = \frac{2x + 2 + 3x}{x(x + 1)} = \frac{5x + 2}{x(x + 1)}$

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

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

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