Answer
$${y^2} + 6y = 2x - 2{x^2} + 352$$
Work Step by Step
$$\eqalign{
& \frac{{dy}}{{dx}} = \frac{{1 - 2x}}{{y + 3}};\,\,\,\,\,\,\,\,\,y\left( 0 \right) = 16 \cr
& {\text{Separating variables leads to}} \cr
& \left( {y + 3} \right)dy = \left( {1 - 2x} \right)dx \cr
& {\text{To solve this equation}}{\text{, determine the antiderivative of each side}} \cr
& \int {\left( {y + 3} \right)dy} = \int {\left( {1 - 2x} \right)dx} \cr
& {\text{integrating by using the power rule}} \cr
& \frac{{{y^2}}}{2} + 3y = x - {x^2} + C\,\,\,\,\,\left( {\bf{1}} \right) \cr
& \cr
& {\text{we can find the constant }}C{\text{ using the initial value problem}}\,\,y\left( 0 \right) = 16 \cr
& y\left( 0 \right) = 16{\text{ implies that }}y = 16{\text{ when }}x = 0 \cr
& {\text{substituting these values into }}\left( {\bf{1}} \right) \cr
& \frac{{{{\left( {16} \right)}^2}}}{2} + 3\left( {16} \right) = \left( 0 \right) - {\left( 0 \right)^2} + C \cr
& 176 = C \cr
& {\text{substitute }}C = 176{\text{ into }}\left( {\bf{1}} \right){\text{ to find the particular solution}} \cr
& \frac{{{y^2}}}{2} + 3y = x - {x^2} + 176 \cr
& {\text{mutliply by 2}} \cr
& {y^2} + 6y = 2x - 2{x^2} + 352 \cr} $$
