I saw this question the first time around missing those parentheses around the [tex]x[/tex], which made the ODE much, much more difficult to solve. This is doable, especially because this ODE is separable.
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\sqrt{-4y+28}[/tex]
[tex]\dfrac{\mathrm dy}{2\sqrt{7-y}}=\mathrm dx[/tex]
[tex]\displaystyle\frac12\int\frac{\mathrm dy}{\sqrt{7-y}}=\int\mathrm dx[/tex]
[tex]-\sqrt{7-y}=x+C[/tex]
[tex]\sqrt{7-y}=-x+C[/tex]
[tex]7-y=(C-x)^2[/tex]
[tex]y=7-(C-x)^2[/tex]
Given that [tex]y(-2)=-2[/tex], we have
[tex]-2=7-(C+2)^2[/tex]
[tex]\implies C=-5,C=1[/tex]
So there are two possible particular solutions:
[tex]\begin{cases}y=-x^2-10x-18\\y=-x^2+2x+6\end{cases}[/tex]
