Answer
$\dfrac{3x^{2}-5x-2}{y^{2}+y-2}\cdot\dfrac{y^{2}+4y-5}{12x^{2}+7x+1}\div\dfrac{5x^{2}-9x-2}{8x^{2}-2x-1}=\dfrac{(y+5)(2x-1)}{(y+2)(5x+1)}$
Work Step by Step
$\dfrac{3x^{2}-5x-2}{y^{2}+y-2}\cdot\dfrac{y^{2}+4y-5}{12x^{2}+7x+1}\div\dfrac{5x^{2}-9x-2}{8x^{2}-2x-1}$
Factor all three rational expressions completely:
$\dfrac{3x^{2}-5x-2}{y^{2}+y-2}\cdot\dfrac{y^{2}+4y-5}{12x^{2}+7x+1}\div\dfrac{5x^{2}-9x-2}{8x^{2}-2x-1}=...$
$...=\dfrac{(3x+1)(x-2)}{(y+2)(y-1)}\cdot\dfrac{(y+5)(y-1)}{(3x+1)(4x+1)}\div\dfrac{(5x+1)(x-2)}{(4x+1)(2x-1)}=...$
Evaluate the operations:
$...=\dfrac{(3x+1)(x-2)(y+5)(y-1)(4x+1)(2x-1)}{(y+2)(y-1)(3x+1)(4x+1)(5x+1)(x-2)}=...$
Simplify by removing the factors that appear both in the numerator and in the denominator:
$...=\dfrac{(y+5)(2x-1)}{(y+2)(5x+1)}$
