Answer
$${\text{ }}F\left( {x,y,z} \right) = x{y^2} + {x^2}y - 10 - z$$
Work Step by Step
$$\eqalign{
& {\text{Let }}z = x{y^2} + {x^2}y - 10 \cr
& {\text{Write in the implicit form }}F\left( {x,y,z} \right) = 0 \cr
& z = x{y^2} + {x^2}y - 10 \cr
& {\text{Rearrange subtracting }}z{\text{ from both sides of the equation}} \cr
& x{y^2} + {x^2}y - 10 - z = 0 \cr
& {\text{We know that }}F\left( {x,y,z} \right) = 0,{\text{ then}} \cr
& {\text{ }}F\left( {x,y,z} \right) = x{y^2} + {x^2}y - 10 - z \cr} $$
