= Solution
Because the original <differential equation> divides by $xy$, its domain requires $x\ne0$ and $y\ne0$. Set $u=y^2$, so $u'=2yy'$. The equation becomes the <first-order linear differential equation>
$$
u'+\frac2x u=-4x-2.
$$
An <integrating factor> is $x^2$, since $(x^2u)'=x^2u'+2xu$. Thus
$$
(x^2u)'=-4x^3-2x^2,\qquad x^2u=-x^4-\frac23x^3+C.
$$
Returning to $y$ gives the explicit general real solutions
$$
\boxed{y(x)=\pm\sqrt{\frac C{x^2}-x^2-\frac23x},\qquad C\in\mathbb R.}
$$
Each sign is fixed on a connected <interval> where $x\ne0$ and the radicand is strictly positive. The strict condition excludes $y=0$, where the original <differential equation> is undefined. There is no additional zero solution.
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