1. We are given the equation of a conic: $$16x^2 + 25y^2 + 160x + 200y + 400 = 0$$. 2. First, group the $x$ and $y$ terms: $$16x^2 + 160x + 25y^2 + 200y + 400 = 0$$. 3. Factor coefficients of squared terms: $$16(x^2 + 10x) + 25(y^2 + 8y) + 400 = 0$$. 4. Complete the square for $x$: Take half of 10, which is 5, square it: $5^2 = 25$. Add and subtract $16 \times 25 = 400$ inside the expression: $$16(x^2 + 10x + 25 - 25) + 25(y^2 + 8y) + 400 = 0$$ $$16((x + 5)^2 - 25) + 25(y^2 + 8y) + 400 = 0$$. 5. Complete the square for $y$: Half of 8 is 4, square it: $4^2 = 16$. Add and subtract $25 \times 16 = 400$: $$16((x + 5)^2 - 25) + 25((y + 4)^2 - 16) + 400 = 0$$. 6. Expand and simplify: $$16(x + 5)^2 - 400 + 25(y + 4)^2 - 400 + 400 = 0$$ $$16(x + 5)^2 + 25(y + 4)^2 - 400 = 0$$. 7. Add 400 to both sides: $$16(x + 5)^2 + 25(y + 4)^2 = 400$$. 8. Divide both sides by 400: $$\frac{16(x + 5)^2}{400} + \frac{25(y + 4)^2}{400} = 1$$ which simplifies to $$\frac{(x + 5)^2}{25} + \frac{(y + 4)^2}{16} = 1$$. This is the equation of an ellipse centered at $(-5, -4)$ with semi-major axis 5 and semi-minor axis 4. **Final answer:** $$\frac{(x + 5)^2}{25} + \frac{(y + 4)^2}{16} = 1$$