Get LCD 
or cross multiply
m-2 times m+1 + m-1 times m+2
divide by m+2 times m+1
m^2-1m-2 + m^2+1m-2
divide by m^2+3m+2
m^2 -4
or m+2 times m-2
divide by m+2 times m+1
cancel m+2
the answer is 
m-2 over m+1
Solve for the roots (x). Given is to be solved as quadratic equation, not as rational expression :-)
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First, simplify the given equation:

m + m -  \frac{2}{m} - \frac{1}{m} + 2 + 1 = 0

2m -   \frac{3}{m} + 3 = 0

2m(m)- \frac{3(m)}{m} + 3(m)=0

2m²+3m-3 = 0

This equation can not be solved using  completing the square and factoring because there are no rational factors.

Using Quadratic Formula the answer is:

x = - \frac{3}{4} + \frac{1}{4}  \sqrt{33}


x = - \frac{3}{4} - \frac{1}{4}  \sqrt{33}

My hand written solution attached.
I hope the solution is clear in your picture viewer :-)