Add and simplify: (x + 5)/(2x − 2) + 3x/(x² + x − 2).

Step-by-step solution

1. 1

   Factor both denominators before choosing a common one. Multiplying the two denominators together would work but produces an unnecessarily large expression that must then be reduced. Factoring reveals the shared piece:

   $2x-2=2(x-1),\qquad x^{2}+x-2=(x+2)(x-1)$

2. 2

   Build the least common denominator. Take each distinct factor to its highest power across the two denominators — the constant $2$, the factor $(x-1)$ that both share, and the factor $(x+2)$:

   $\text{LCD}=2(x-1)(x+2)$

3. 3

   Rewrite each fraction over the LCD. The first needs an extra $(x+2)$; the second needs an extra $2$:

   $\frac{(x+5)(x+2)}{2(x-1)(x+2)}+\frac{2\cdot 3x}{2(x-1)(x+2)}$

4. 4

   Add the numerators and collect.

   $(x+5)(x+2)=x^{2}+7x+10$

   $x^{2}+7x+10+6x=x^{2}+13x+10$

   so the sum is

   $\frac{x^{2}+13x+10}{2(x-1)(x+2)}$

5. 5

   Check whether the result reduces. A cancellation would need $x-1$ or $x+2$ to divide $x^2+13x+10$. At $x=1$ the numerator is $24\neq 0$; at $x=-2$ it is $4-26+10=-12\neq 0$. Neither is a factor, so the expression is already in lowest terms.

   $\boxed{\dfrac{x^{2}+13x+10}{2(x-1)(x+2)}}\qquad (x\neq 1,\ x\neq -2)$

6. 6

   Verify with a test value. At $x=3$: the original is $\tfrac{8}{4}+\tfrac{9}{10}=2+0.9=2.9$, and the answer gives $\tfrac{9+39+10}{2(2)(5)}=\tfrac{58}{20}=2.9$. They agree.