Question 203065

{{{((n^2+5n+6)/(n^2+7n+12))((n^2+9n+20)/(n^2+11n+30))}}} Start with the given expression.



{{{(((n+3)(n+2))/(n^2+7n+12))((n^2+9n+20)/(n^2+11n+30))}}} Factor {{{n^2+5n+6}}} to get {{{(n+3)(n+2)}}}.



{{{(((n+3)(n+2))/((n+4)(n+3)))((n^2+9n+20)/(n^2+11n+30))}}} Factor {{{n^2+7n+12}}} to get {{{(n+4)(n+3)}}}.



{{{(((n+3)(n+2))/((n+4)(n+3)))(((n+5)(n+4))/(n^2+11n+30))}}} Factor {{{n^2+9n+20}}} to get {{{(n+5)(n+4)}}}.



{{{(((n+3)(n+2))/((n+4)(n+3)))(((n+5)(n+4))/((n+6)(n+5)))}}} Factor {{{n^2+11n+30}}} to get {{{(n+6)(n+5)}}}.



{{{((n+3)(n+2)(n+5)(n+4))/((n+4)(n+3)(n+6)(n+5))}}} Combine the fractions. 



{{{(highlight((n+3))(n+2)highlight((n+5))highlight((n+4)))/(highlight((n+4))highlight((n+3))(n+6)highlight((n+5)))}}} Highlight the common terms. 



{{{(cross((n+3))(n+2)cross((n+5))cross((n+4)))/(cross((n+4))cross((n+3))(n+6)cross((n+5)))}}} Cancel out the common terms. 



{{{(n+2)/(n+6)}}} Simplify. 



So {{{((n^2+5n+6)/(n^2+7n+12))((n^2+9n+20)/(n^2+11n+30))}}} simplifies to {{{(n+2)/(n+6)}}}.



In other words, {{{((n^2+5n+6)/(n^2+7n+12))((n^2+9n+20)/(n^2+11n+30))=(n+2)/(n+6)}}}