(c) Write as a single fraction in its simplest form. \[ \frac{2 x+4}{x+1}-\frac{x}{x-3} \]

To combine the fractions \( \frac{2x+4}{x+1} - \frac{x}{x-3} \), we first find a common denominator, which is \( (x + 1)(x - 3) \). The first fraction becomes: \[ \frac{(2x + 4)(x - 3)}{(x + 1)(x - 3)} \] And the second fraction becomes: \[ \frac{x(x + 1)}{(x + 1)(x - 3)} \] Now we can combine the fractions: \[ \frac{(2x + 4)(x - 3) - x(x + 1)}{(x + 1)(x - 3)} \] Next, we simplify the numerator: 1. Expand \( (2x + 4)(x - 3) \): \[ 2x^2 - 6x + 4x - 12 = 2x^2 - 2x - 12 \] 2. Expand \( -x(x + 1) \): \[ -x^2 - x \] Now combine: \[ 2x^2 - 2x - 12 - x^2 - x = (2x^2 - x^2) + (-2x - x) - 12 = x^2 - 3x - 12 \] So, the combined fraction is: \[ \frac{x^2 - 3x - 12}{(x + 1)(x - 3)} \] Next, we factor the numerator \( x^2 - 3x - 12 \): \[ x^2 - 3x - 12 = (x - 6)(x + 2) \] Thus, the expression becomes: \[ \frac{(x - 6)(x + 2)}{(x + 1)(x - 3)} \] This is the simplest form of the original expression: \[ \frac{(x - 6)(x + 2)}{(x + 1)(x - 3)} \]