1. **State the problem:** We need to graph the ellipse given by the equation $$16x^2 + 25y^2 + 64x + 50y - 311 = 0$$ and find its center, axes lengths, and intercepts. 2. **Rewrite the equation grouping x and y terms:** $$16x^2 + 64x + 25y^2 + 50y = 311$$ 3. **Complete the square for x and y terms:** - For x: factor out 16: $$16(x^2 + 4x)$$ - For y: factor out 25: $$25(y^2 + 2y)$$ 4. **Complete the square inside each parenthesis:** - For $$x^2 + 4x$$, add and subtract $$4$$ (since $$(\frac{4}{2})^2 = 4$$): $$x^2 + 4x + 4 - 4 = (x+2)^2 - 4$$ - For $$y^2 + 2y$$, add and subtract $$1$$ (since $$(\frac{2}{2})^2 = 1$$): $$y^2 + 2y + 1 - 1 = (y+1)^2 - 1$$ 5. **Substitute back and simplify:** $$16((x+2)^2 - 4) + 25((y+1)^2 - 1) = 311$$ $$16(x+2)^2 - 64 + 25(y+1)^2 - 25 = 311$$ $$16(x+2)^2 + 25(y+1)^2 = 311 + 64 + 25$$ $$16(x+2)^2 + 25(y+1)^2 = 400$$ 6. **Divide both sides by 400 to get the standard ellipse form:** $$\frac{(x+2)^2}{\frac{400}{16}} + \frac{(y+1)^2}{\frac{400}{25}} = 1$$ $$\frac{(x+2)^2}{25} + \frac{(y+1)^2}{16} = 1$$ 7. **Interpret the ellipse parameters:** - Center: $$(-2, -1)$$ - Semi-major axis $$a = 5$$ (since $$\sqrt{25} = 5$$) - Semi-minor axis $$b = 4$$ (since $$\sqrt{16} = 4$$) 8. **Summary:** The ellipse is centered at $$(-2, -1)$$ with horizontal semi-major axis length 5 and vertical semi-minor axis length 4. **Final equation for graphing:** $$\frac{(x+2)^2}{25} + \frac{(y+1)^2}{16} = 1$$