The given problem asks to write the expression

$\frac{2x + 4}{x + 1} - \frac{x}{x - 3}$

as a single fraction in its simplest form.

Step 1: Find a Common Denominator

The common denominator for the fractions $\frac{2x + 4}{x + 1}$ and $\frac{x}{x - 3}$ is $(x + 1)(x - 3)$.

Step 2: Rewrite Each Fraction with the Common Denominator

Rewrite each fraction using the common denominator:

$\frac{2x + 4}{x + 1} = \frac{(2x + 4)(x - 3)}{(x + 1)(x - 3)}$

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

Step 3: Subtract the Fractions

Combine the fractions:

$\frac{(2x + 4)(x - 3) - x(x + 1)}{(x + 1)(x - 3)}$

Step 4: Simplify the Numerator

Expand the numerator:

1. Expand $(2x + 4)(x - 3)$:

   $(2x + 4)(x - 3) = 2x^2 - 6x + 4x - 12 = 2x^2 - 2x - 12$

2. Expand $x(x + 1)$:

   $x(x + 1) = x^2 + x$

3. Subtract the second expansion from the first:

   $(2x^2 - 2x - 12) - (x^2 + x) = 2x^2 - 2x - 12 - x^2 - x = x^2 - 3x - 12$

Step 5: Combine into a Single Fraction

The simplified expression is:

$\frac{x^2 - 3x - 12}{(x + 1)(x - 3)}$

Step 6: Factor the Numerator (if possible)

Factor the numerator $x^2 - 3x - 12$:

$x^2 - 3x - 12 = (x - 6)(x + 2)$

So, the final simplified form is:

$\frac{(x - 6)(x + 2)}{(x + 1)(x - 3)}$

Final Answer

$\frac{(x - 6)(x + 2)}{(x + 1)(x - 3)}$

Would you like more details or have any questions?

Further Related Questions:

1. How do you find a common denominator for rational expressions?
2. How can you factor quadratic expressions?
3. What are some common mistakes when simplifying complex fractions?
4. How do you handle fractions with variables in the denominator?
5. How can you verify that your simplified expression is correct?

Tip:

Always check for factorization opportunities in both the numerator and the denominator after simplifying complex fractions!