Question 178214
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*[tex \LARGE \frac {\frac {2x^2 + 7x + 3}{x^2 + 3x - 4}}{\frac {x^2 + 7x + 12}{2x^2 - 3x + 1}}]


First thing is to factor each of the polynomial expressions:


*[tex \Large 2x^2 + 7x + 3 = (2x + 1)(x + 3)]


*[tex \Large x^2 + 3x - 4 = (x + 4)(x - 1) ]


*[tex \Large x^2 + 7x + 12 = (x + 3)(x + 4) ]


*[tex \Large 2x^2 - 3x + 1 = (2x - 1)(x - 1) ]


So the expression becomes:


*[tex \LARGE \frac {\frac { (2x + 1)(x + 3) }{ (x + 4)(x - 1) }}{\frac { (x + 3)(x + 4) }{ (2x - 1)(x - 1) }}]


Now invert and multiply:


*[tex \LARGE \left(\frac { (2x + 1)(x + 3) }{ (x + 4)(x - 1) }\right) \left({\frac { (2x - 1)(x - 1) }{  (x + 3)(x + 4) } \right) = \frac { (2x + 1)(x + 3) (2x - 1)(x - 1)}{ (x + 4)(x - 1)(x + 3)(x + 4) } ]


Eliminate factors common to both numerator and denominator:


*[tex \LARGE \frac { (2x + 1)(2x - 1)  }{ (x + 4) (x + 4)} ]


The numerator binomials are a conjugate pair, so the product is the difference of 2 squares, and the product of the denominator can be found by applying FOIL:



*[tex \LARGE \frac { 4x^2 - 1  }{ x^2 + 8x + 16 } ]






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