Answer is C.
Finding the factors of 1st term – $4x^{^{3}}+3x^{2}y-9xy^{2}+2y^{3}$
$4x^{^{3}}-4x^{2}y+7x^{2}y-7xy^{2}-2xy^{2}+2y^{3}$
$4x^{2}(x-y)+7xy(x-y)-2y^{2}(x-y)$
$(4x^{2}+7xy-2y^{2})(x-y)$
$(4x^{2}+8xy-xy-2y^{2})(x-y)$
$4x(x+2y)-y(x+2y)$
Factors are $(4x-y)(x+2y)$
Finding the factors of 2nd term – $x^{2}+xy-2y^{2}$
$x^{2}-xy+2xy-2y^{2}$
$x(x-y)+2y(x-y)$
Factors are $(x+2y)(x-y)$
So, HCF is the common factors between 1st term and 2nd term
$(x+2y)(x-y)$