1. (a) Simplify \( \frac{3x^3 - x - 2}{x^2 - 1} \) - Edexcel - A-Level Maths Pure - Question 3 - 2006 - Paper 4

We start with the previous result:

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

We also have to add the term ( \frac{1}{x(x + 1)} ). To combine these into a single fraction, we find a common denominator, which will be (x(x + 1)). Thus, re-writing the first term:

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

Now, we can express the overall fraction as:

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

To simplify further, we check for common factors or any possible cancellations. In the numerator, we can factor:

$3x^2 + 2x - 1 = (3x - 1)(x + 1)$

Thus, combining this, we obtain:

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

Finally, we cancel the (x + 1):

$\frac{3x - 1}{x}$