Simplify (2x − 4 + (x + 1)(x − 2)) / ((x + 1)(x − 2)).

Step-by-step solution

1. 1

   Do not cancel before the numerator is a single product. The numerator is a sum, and only common factors may be cancelled. So the first job is to expand and re-factor the top; leaving $(x+1)(x-2)$ visible on top and cancelling it against the bottom would be wrong, because the $2x-4$ term is still attached by addition.

2. 2

   Expand the product in the numerator.

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

   so the numerator becomes

   $2x-4+x^2-x-2=x^2+x-6$

3. 3

   Factor the collected numerator. Two numbers multiplying to $-6$ and adding to $+1$ are $3$ and $-2$:

   $x^2+x-6=(x+3)(x-2)$

4. 4

   Cancel the common factor.

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

   $\boxed{\dfrac{x+3}{x+1}}$

5. 5

   State the restrictions and spot-check. The cancellation is valid only where $x\neq 2$, and the original expression also requires $x\neq -1$; both exclusions survive into the answer's domain. Testing $x=5$: the original numerator is $10-4+(6)(3)=24$ and the denominator is $(6)(3)=18$, giving $\tfrac{24}{18}=\tfrac{4}{3}$, while $\tfrac{5+3}{5+1}=\tfrac{8}{6}=\tfrac{4}{3}$ — they agree.

6. 6

   Note the second expression on the same worksheet. $\dfrac{6+11(x-1)}{x+3}=\dfrac{11x-5}{x+3}$ after expanding $6+11x-11$; since $11x-5$ has no factor of $x+3$ (it is nonzero at $x=-3$), that one is already in lowest terms.