One awkward fraction is really two simple ones.

Add two fractions and you get one. Partial fractions run that backwards, and the picture shows why it has to work.

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This is $y = \dfrac{5x + 1}{(x - 1)(x + 2)}$. It has three branches and shoots off to infinity at $x = 1$ and $x = -2$, where the denominator is zero.

Now draw $\dfrac{2}{x - 1}$ and $\dfrac{3}{x + 2}$. Each has only one of the two asymptotes. Near $x = 1$ the curve follows the first; near $x = -2$ it follows the second.

At $x = 3$ the simple curves have heights $1$ and $0.6$, and $1 + 0.6 = 1.6$ lands exactly on the curve. It happens at every $x$:

A shortcut for the same thing: cover up $(x - 1)$ in the fraction and put $x = 1$ into what is left, giving $A = \tfrac{5 + 1}{1 + 2} = 2$. Cover up $(x + 2)$ and put $x = -2$ in to get $B = 3$.

Split it, and each piece is easy.

Repeated factors, improper fractions, the algebra of fractions, and twelve graded exam questions with full worked solutions are waiting. Chapters 7 and 15 lean on this: the binomial series and integration.

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