[tex] \int \frac{2x + 4}{x {}^{2}.(x - 2) } dx \\ \\ \frac{2x + 4}{x {}^{2} .(x - 2)} = \frac{a}{x} + \frac{b}{x {}^{2} } + \frac{c}{x - 2} \\ \\ \frac{2x + 4}{x {}^{2}.(x - 2) } = \frac{ax {}^{2} + bx}{x {}^{3} } + \frac{c}{x - 2} \\ \\ \frac{2x + 4}{x {}^{2} .(x - 2)} = \frac{(ax^{2} + bx).(x - 2) + cx {}^{3} }{x {}^{3} .(x - 2)} \\ \\ \frac{2x + 4}{x {}^{2}.(x - 2) } = \frac{ax {}^{3} - 2ax {}^{2} + bx {}^{2} - 2bx + cx {}^{3} }{x {}^{3}.(x - 2) } \\ \\ \frac{2x + 4}{x {}^{2}.(x - 2) } = \frac{x.(ax^{2} - 2ax + bx - 2b + cx {}^{2} )}{x {}^{3} .(x - 2)} \\ \\ 2x + 4 = ax^{2} - 2ax + bx - 2b + cx {}^{2}[/tex]

[tex]0x {}^{2} + 2x + 4 = ax^{2} - 2ax + bx - 2b + cx {}^{2} \\ \\ \begin{cases}a + c = 0 \: \: \to \: c = 2 \\ - 2a + b = 2 \: \: \to \: \: a = - 2\\ - 2b = 4 \: \to \: \: b = - 2 \end{cases} \\ \\ [/tex]

[tex] \int \frac{2x + 4}{x {}^{2}.(x - 2) } dx \: \: \to \: \: \int \left( - \frac{2}{x} - \frac{2}{x {}^{2} } + \frac{2}{x - 2} \right)dx \\ \\ - \int \frac{2}{x} dx - \int \frac{2}{x {}^{2} } dx + \int \frac{2}{x - 2} dx \\ \\ - 2 \int \frac{1}{x} dx - 2 \int \frac{1}{x {}^{2} } + 2 \int \frac{1}{x - 2} dx \\ \\ \boxed{- 2 \ln( |x| ) + \frac{2}{x} + 2 \ln( |x - 2| ) + c}[/tex]