Question 888902
{{{(q^3+4q^2-5q)/(q^2-2q+1)}}}÷{{{(q^2+q-2)/(q^4-8q)}}}*{{{(q-4)/(q^2-2q+4)}}}<br>

First we need to remember that a rational expression divided by a rational expression is the same as a rational expression times the other expressions reciprocal.  So:<br>

{{{(q^3+4q^2-5q)/(q^2-2q+1)}}}÷{{{(q^2+q-2)/(q^4-8q)}}}*{{{(q-4)/(q^2-2q+4)}}}
={{{(q^3+4q^2-5q)/(q^2-2q+1)}}}*{{{(q^4-8q)/(q^2+q-2)}}}*{{{(q-4)/(q^2-2q+4)}}}<br>

Now we Factor all the parts of this expression to see if anything cancels out.

{{{(q^3+4q^2-5q)/(q^2-2q+1)}}}*{{{(q^4-8q)/(q^2+q-2)}}}*{{{(q-4)/(q^2-2q+4)}}}
={{{(q(q^2+4q-5))/((q-1)^2)}}}*{{{(q(q^3-8))/((q-2)(q+1))}}}*{{{(q-4)/(q^2-2q+4)}}}
={{{(q((q-1)(q+5)))/((q-1)^2)}}}*{{{(q(q-2)(q^2+2q+4))/((q-2)(q+1))}}}*{{{(q-4)/(q^2-2q+4)}}}<br>

Now we see that the {{{(q-1)}}} terms in the first expression cancel out as well as the {{{(q-2)}}} terms in the second expression.  The {{{(q^2+2q+4)}}} terms also cancel out so we are left with:<br>

={{{(q(q+5)))/(q-1)}}}*{{{q/(q+1))}}}*{{{(q-4)/1}}}