To solve it, we can use the method of separation of variables. We can rewrite the equation as:
$$\frac{dx}{x^2 + 4x + y^2 + 1} = -\frac{dy}{2y(y+2)}$$
Now, we can integrate both sides:
$$\int \frac{dx}{x^2 + 4x + y^2 + 1} = -\int \frac{dy}{2y(y+2)}$$
Let's
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