Express \[ \frac{3x + 5}{x^2 + x - 12} - \frac{2}{x - 3} \] as a single fraction in its simplest form. - Edexcel - A-Level Maths Pure - Question 3 - 2013 - Paper 8

To combine the fractions, we need a common denominator. The common denominator is ((x + 4)(x - 3)). Therefore:

[ \frac{3x + 5}{(x + 4)(x - 3)} - \frac{2(x + 4)}{(x - 3)(x + 4)} = \frac{3x + 5 - 2(x + 4)}{(x + 4)(x - 3)} ]

Simplifying the numerator:

[ 3x + 5 - 2(x + 4) = 3x + 5 - 2x - 8 = x - 3 ]

So we have:

[ \frac{x - 3}{(x + 4)(x - 3)} ]

The (x - 3) terms in the numerator and denominator cancel out:

[ \frac{1}{x + 4} ]

Thus, the final answer is:

[ \frac{1}{x + 4} ]