Aus Online Mathematik Brückenkurs 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}\centerdot \frac{a+1}{a+1}-\frac{ax^{2}}{\left( a+1 \right)^{2}}=\frac{ax\left( a+1 \right)-ax^{2}}{\left( a+1 \right)^{2}}

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

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