Answer
$\dfrac{(x+2)(5y+1)}{(x+5)(2y-1)}$
Work Step by Step
Factoring the expressions and cancelling the common factors between the numerator and the denominator, the given expression, $
\dfrac{x^2+x-2}{3y^2-5y-2}\cdot\dfrac{12y^2+y-1}{x^2+4x-5}\div\dfrac{8y^2-6y+1}{5y^2-9y-2}
,$ simplifies to
\begin{array}{l}\require{cancel}
\dfrac{x^2+x-2}{3y^2-5y-2}\cdot\dfrac{12y^2+y-1}{x^2+4x-5}\cdot\dfrac{5y^2-9y-2}{8y^2-6y+1}
\\\\
\dfrac{(x+2)(x-1)}{(3y+1)(y-2)}\cdot\dfrac{(4y-1)(3y+1)}{(x+5)(x-1)}\cdot\dfrac{(5y+1)(y-2)}{(4y-1)(2y-1)}
\\\\
\dfrac{(x+2)(\cancel{x-1})}{(\cancel{3y+1})(\cancel{y-2})}\cdot\dfrac{(\cancel{4y-1})(\cancel{3y+1})}{(x+5)(\cancel{x-1})}\cdot\dfrac{(5y+1)(\cancel{y-2})}{(\cancel{4y-1})(2y-1)}
\\\\=
\dfrac{(x+2)(5y+1)}{(x+5)(2y-1)}
.\end{array}
