Express \[ \frac{2x^2 + 3x}{(2x + 3)(x - 2)} \div \frac{6}{x^2 - x - 2} \] as a single fraction in its simplest form. - Edexcel - A-Level Maths Pure - Question 4 - 2006 - Paper 5

To express this as a single fraction, we first need to find the reciprocal of the second fraction and convert the division into multiplication:

[ \frac{2x^2 + 3x}{(2x + 3)(x - 2)} \times \frac{x^2 - x - 2}{6} ]

Next, we factor the denominator of the second fraction:

[ x^2 - x - 2 = (x - 2)(x + 1) ]

Updating the expression gives us:

[ \frac{2x^2 + 3x}{(2x + 3)(x - 2)} \times \frac{(x - 2)(x + 1)}{6} ]

Now we can simplify by canceling the common factor (x - 2):

[ \frac{2x^2 + 3x (x + 1)}{(2x + 3) \cdot 6} ]

Next, we simplify the numerator:

[ 2x^2 + 3x = x(2x + 3) \Rightarrow \frac{x(2x + 3)(x + 1)}{(2x + 3) \cdot 6} ]

At this point, we can cancel (2x + 3) from both numerator and denominator:

[ \frac{x(x + 1)}{6} ]

This is the simplest form of the expression.