1. **State the problem:** Solve the equation $x(x+2) + y(y+2) = 2000$ for possible values of $x$ and $y$. 2. **Rewrite the equation:** Expand the terms: $$x^2 + 2x + y^2 + 2y = 2000$$ 3. **Complete the square:** Group $x$ and $y$ terms: $$x^2 + 2x + y^2 + 2y = 2000$$ Add and subtract 1 for both $x$ and $y$ to complete the square: $$ (x^2 + 2x + 1) + (y^2 + 2y + 1) = 2000 + 1 + 1 $$ This simplifies to: $$ (x+1)^2 + (y+1)^2 = 2002 $$ 4. **Interpretation:** The equation represents a circle centered at $(-1, -1)$ with radius $\sqrt{2002}$. 5. **Solution set:** All points $(x,y)$ satisfying: $$ (x+1)^2 + (y+1)^2 = 2002 $$ are solutions to the original equation. **Final answer:** $$ (x+1)^2 + (y+1)^2 = 2002 $$