21 Write as a single fraction in its simplest form - OCR - GCSE Maths - Question 21 - 2020 - Paper 6

We will rewrite each fraction with the common denominator:

1. ( \frac{x}{x + 1} = \frac{x(x + 2)(x - 2)}{(x + 1)(x + 2)(x - 2)} )
2. ( \frac{6x}{x + 2} = \frac{6x(x + 1)(x - 2)}{(x + 1)(x + 2)(x - 2)} )
3. ( \frac{x - 2}{x^2 - 4} = \frac{x - 2}{(x - 2)(x + 2)} = \frac{1}{x + 2} ) so we rewrite it as ( \frac{(x + 1)(x - 2)}{(x + 1)(x + 2)(x - 2)} ).

Now we can add and subtract the numerators: [ \frac{x(x + 2)(x - 2) + 6x(x + 1)(x - 2) - (x + 1)(x - 2)}{(x + 1)(x + 2)(x - 2)} ]

Expand and simplify the numerator:

1. ( x(x + 2)(x - 2) = x(x^2 - 4) = x^3 - 4x )
2. ( 6x(x + 1)(x - 2) = 6x(x^2 - x - 2) = 6x^3 - 6x^2 - 12x )
3. Combine all these terms and simplify.

Finally, combine the simplified numerator with the common denominator: [ \frac{x^3 + 6x^3 - 4x - 6x^2 - 12x + x + 2}{(x + 1)(x + 2)(x - 2)} ] Combine like terms to get your final answer in simplest form.