1. **State the problem:** Determine if the line $y=8-x$ is tangent to the curve defined by $2x^2 + xy = -16$. 2. **Substitute the line equation into the curve:** Replace $y$ in the curve equation with $8-x$: $$2x^2 + x(8 - x) = -16$$ 3. **Simplify the equation:** $$2x^2 + 8x - x^2 = -16$$ $$x^2 + 8x = -16$$ 4. **Bring all terms to one side:** $$x^2 + 8x + 16 = 0$$ 5. **Check the discriminant to determine tangency:** The quadratic is $x^2 + 8x + 16 = 0$. The discriminant $\Delta = b^2 - 4ac = 8^2 - 4 \times 1 \times 16 = 64 - 64 = 0$. 6. **Interpretation:** A discriminant of zero means the quadratic has exactly one real root, so the line touches the curve at exactly one point. **Conclusion:** The line $y=8-x$ is tangent to the curve $2x^2 + xy = -16$ because the substitution leads to a quadratic with a single root (discriminant zero).