<s>[INST] Splitting fractions with a linear denominator: $\frac{2x-1}{x+2}$ How can $$\frac{2x-1}{x+2}$$ be split to give $$A-\frac{B}{x+2}$$
where $A$ and $B$ are integers?
The solution is $$2-\frac{5}{x+2}.$$
 [/INST] $\dfrac{2x-1}{x+2}:$
\begin{array}{ccccccc}
    &&&& 2 \\
    &&&& -- & -- & --\\
    x & + & 2 & | & 2x & - & 1 \\
    &&&&2x & + & 4 \\
    &&&& -- & -- & --\\
    &&&&&&-5
\end{array}
So $\dfrac{2x-1}{x+2} = 2 + \dfrac{-5}{x+2} = 2 - \dfrac{5}{x+2}$
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