Answer
$14$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {5, - 2} \right)} \left( {{x^2}y + 3x{y^2} + 4} \right) \cr
& {\text{The function }}f\left( {x,y} \right) = {x^2}y + 3x{y^2} + 4{\text{ is a polynomial}}{\text{, so we}} \cr
& {\text{can find the limit by direct substitution}}{\text{.}} \cr
& {\text{Substitute 5 for }}x{\text{ and }} - 2{\text{ for }}y \cr
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {5, - 2} \right)} \left( {{x^2}y + 3x{y^2} + 4} \right) = {\left( 5 \right)^2}\left( { - 2} \right) + 3\left( 5 \right){\left( { - 2} \right)^2} + 4 \cr
& = 25\left( { - 2} \right) + 3\left( 5 \right)\left( 4 \right) + 4 \cr
& = - 50 + 60 + 4 \cr
& = 14 \cr} $$