Question 119220
{{{((14x-7)/(x^2+3x-4)*(x^2+6x+8)/(2x^2+5x-3))/((x^2+2x)/(x^2+2x-3))}}}



Let's simplify the numerator {{{(14x-7)/(x^2+3x-4)*(x^2+6x+8)/(2x^2+5x-3)}}}





{{{((14x-7)/(x^2+3x-4))((x^2+6x+8)/(2x^2+5x-3))}}} Start with the given expression


{{{((7(2x-1))/(x^2+3x-4))((x^2+6x+8)/(2x^2+5x-3))}}}   Factor {{{14x-7}}} to get {{{7(2x-1)}}} 


{{{((7(2x-1))/((x+4)(x-1)))((x^2+6x+8)/(2x^2+5x-3))}}}   Factor {{{x^2+3x-4}}} to get {{{(x+4)(x-1)}}} 


{{{((7(2x-1))/((x+4)(x-1)))(((x+4)(x+2))/(2x^2+5x-3))}}}   Factor {{{x^2+6x+8}}} to get {{{(x+4)(x+2)}}} 


{{{((7(2x-1))/((x+4)(x-1)))(((x+4)(x+2))/((x+3)(2x-1)))}}}   Factor {{{2x^2+5x-3}}} to get {{{(x+3)(2x-1)}}} 



{{{7(2x-1)(x+4)(x+2)/(x+4)(x-1)(x+3)(2x-1)}}} Combine the fractions



{{{7cross((2x-1))cross((x+4))(x+2)/cross((x+4))(x-1)(x+3)cross((2x-1))}}} Cancel like terms



{{{7(x+2)/(x-1)(x+3)}}} Simplify






So {{{((14x-7)/(x^2+3x-4))((x^2+6x+8)/(2x^2+5x-3))}}} simplifies to {{{7(x+2)/(x-1)(x+3)}}}. In other words {{{((14x-7)/(x^2+3x-4))((x^2+6x+8)/(2x^2+5x-3))=7(x+2)/(x-1)(x+3)}}}



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So {{{((14x-7)/(x^2+3x-4)*(x^2+6x+8)/(2x^2+5x-3))/((x^2+2x)/(x^2+2x-3))}}} becomes



{{{(7(x+2)/(x-1)(x+3))/((x^2+2x)/(x^2+2x-3))}}}



{{{(7(x+2)/(x-1)(x+3))*((x^2+2x-3)/(x^2+2x))}}} Multiply the first fraction by the reciprocal of the second fraction




{{{((7(x+2))/(x-1)(x+3))(((x+3)(x-1))/(x^2+2x))}}}   Factor {{{x^2+2x-3}}} to get {{{(x+3)(x-1)}}} 


{{{((7(x+2))/(x-1)(x+3))(((x+3)(x-1))/(x(x+2)))}}}   Factor {{{x^2+2x}}} to get {{{x(x+2)}}} 



{{{7(x+2)(x+3)(x-1)/(x-1)(x+3)x(x+2)}}} Combine the fractions




{{{7cross((x+2))cross((x+3))cross((x-1))/cross((x-1))cross((x+3))x*cross((x+2))}}} Cancel like terms



{{{7/x}}} Simplify







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Answer:




So {{{((14x-7)/(x^2+3x-4)*(x^2+6x+8)/(2x^2+5x-3))/((x^2+2x)/(x^2+2x-3))}}} simplifies to {{{7/x}}}




In other words, 


{{{((14x-7)/(x^2+3x-4)*(x^2+6x+8)/(2x^2+5x-3))/((x^2+2x)/(x^2+2x-3))=7/x}}}