From Förberedande kurs i matematik 1

We multiply the top and bottom of the first term by \displaystyle a+1, so that both terms then have the same denominator,

\displaystyle \frac{ax}{a+1}\cdot\frac{a+1}{a+1} - \frac{ax^{2}}{(a+1)^{2}} = \frac{ax(a+1)-ax^{2}}{(a+1)^{2}}\,\textrm{.}

Because both terms in the numerator contain the factor \displaystyle ax, we take out that factor, obtaining

\displaystyle \frac{ax(a+1-x)}{(a+1)^{2}}

and see that the answer cannot be simplified any further.

Note: It is only factors in the numerator and denominator that can cancel each other out, not individual terms. Hence, the following "cancellation" is wrong

\displaystyle \frac{ax(\rlap{/\,/\,/\,/}a+1)-ax^2}{(a+1)^{2\llap{/}}} = \frac{ax-ax^{2}}{a+1}\,\textrm{.}