Factor the following completely. \( -50 x^{3} y^{2}+90 x^{2} y^{3}+120 x^{3} y^{3} \)

To factor the expression completely, let's start by identifying the greatest common factor (GCF) of the terms in \( -50 x^{3} y^{2} + 90 x^{2} y^{3} + 120 x^{3} y^{3} \). The GCF of the coefficients is 10, and for the variables, the smallest powers are \( x^{2} \) and \( y^{2} \). Thus, the GCF is \( 10 x^{2} y^{2} \). Now, we can factor this out: \[ -50 x^{3} y^{2} + 90 x^{2} y^{3} + 120 x^{3} y^{3} = 10 x^{2} y^{2} (-5 x + 9 y + 12 x y). \] Next, we can attempt to factor the trinomial \( -5 x + 9 y + 12 x y \). Reorganizing it gives us \( 12 x y - 5 x + 9 y \), which can be rearranged to factor further: \[ = 10 x^{2} y^{2} (12 x y - 5 x + 9 y). \] After checking, further factoring appears difficult, leading us back to the factored form. Thus, the complete factoring results in: \[ 10 x^{2} y^{2} (12 x y - 5 x + 9 y). \]