What are the steps to rewrite #(5x+3)/(x^(2)+4x+7# so it end up #((5(2x+4))/(2(x^(2)+4x+7)))-(7)/((x^(2)+4x+7))# ?

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Answer 1 · 2018-03-01T14:50:19

As proved below.

Given #(5x + 3) / (x^2 + 4x + 7)#

Multiply and divide by #color(brown)(2#

#=> ((5x + 3) * color(brown)(2)) / ((x^2 + 4x + 7) *color(brown)(2))#

#=> (10x + 6) / (2 * (x^2 + 4x + 7) )#

Add and subtract #color(blue)(14)#

#=> (10x + 6 + color(blue)(14 - 14)) / (2 * (x^2 + 4x + 7) )#

#=> (10x + 20) / (2*(x^2 + 4x + 7)) - cancel(14)^color(red)7 / (cancel2*(x^2 + 4x + 7) )#

#=> (5(2 x+ 4)) / (2(x^2 + 4x + 7)) - 7 / (x^2 + 4x + 7) #

Hence proved.