Simplify the rational expression (x^2 + 3x - 10) / (x^2 - 4) and state the restrictions on x.

Step-by-step solution

1. 1

   Factor the numerator by finding a pair with the right sum and product. For $x^2 + 3x - 10$ you need two numbers multiplying to $-10$ and adding to $3$: those are $5$ and $-2$.

   $x^2 + 3x - 10 = (x+5)(x-2)$

2. 2

   Factor the denominator as a difference of squares. $x^2 - 4 = x^2 - 2^2$, so

   $x^2 - 4 = (x-2)(x+2)$

   The pattern $a^2 - b^2 = (a-b)(a+b)$ is worth spotting first — it is faster than searching for a factor pair.

3. 3

   Record the restrictions before cancelling. The original expression is undefined wherever the denominator vanishes, so $x \neq 2$ and $x \neq -2$. This must be written down now, because the next step erases the evidence for $x = 2$.

4. 4

   Cancel the common factor. Both top and bottom carry $(x-2)$:

   $\frac{(x+5)(x-2)}{(x-2)(x+2)} = \frac{x+5}{x+2}$

   The cancelled $(x-2)$ leaves a removable hole in the graph at $x = 2$ — the simplified form has a value there, $\tfrac{7}{4}$, but the original does not.

5. 5

   Verify with a test value. At $x = 3$ the original is $\tfrac{9+9-10}{9-4} = \tfrac{8}{5} = 1.6$, and the simplified form gives $\tfrac{3+5}{3+2} = \tfrac{8}{5} = 1.6$. The two agree everywhere except at the excluded points.