\(\frac{x+2}{x+3}-\frac{5}{x^2+x-6}+\frac{1}{2-x}=\frac{x+2}{x+3}-\frac{5}{x^2+3x-2x-6}-\frac{1}{x-2}\)\(=\frac{x+2}{x+3}-\frac{5}{x\left(x+3\right)-2\left(x+3\right)}-\frac{1}{x-2}=\frac{x+2}{x+3}-\frac{5}{\left(x+3\right)\left(x-2\right)}-\frac{1}{x-2}\)
\(=\frac{\left(x+2\right)\left(x-2\right)-5-1.\left(x+3\right)}{\left(x+3\right)\left(x-2\right)}=\frac{x^2-4-5-x-3}{\left(x+3\right)\left(x-2\right)}=\frac{x^2-x-12}{\left(x+3\right)\left(x-2\right)}\)
\(=\frac{x^2-4x+3x-12}{\left(x+3\right)\left(x-2\right)}=\frac{x\left(x-4\right)+3\left(x-4\right)}{\left(x+3\right)\left(x-2\right)}=\frac{\left(x-4\right)\left(x+3\right)}{\left(x-3\right)\left(x-2\right)}=\frac{x-4}{x-2}\)
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