1. **State the problem:** Solve the equation $$y^2 + 2y - x^2 + 6y = -10x$$ for $y$ in terms of $x$. 2. **Combine like terms:** Group the $y$ terms together: $$y^2 + (2y + 6y) - x^2 = -10x$$ which simplifies to $$y^2 + 8y - x^2 = -10x$$ 3. **Rewrite the equation:** Move all terms to one side: $$y^2 + 8y = x^2 - 10x$$ 4. **Complete the square for $y$:** Recall the formula for completing the square: $$y^2 + 2ay = (y + a)^2 - a^2$$ Here, $8y = 2 imes 4 imes y$, so $a=4$. Add and subtract $4^2 = 16$ on the left side: $$y^2 + 8y + 16 - 16 = x^2 - 10x$$ which is $$(y + 4)^2 - 16 = x^2 - 10x$$ 5. **Isolate the perfect square:** $$(y + 4)^2 = x^2 - 10x + 16$$ 6. **Complete the square for $x$ on the right side:** $$x^2 - 10x + 16 = (x^2 - 10x + 25) - 9 = (x - 5)^2 - 9$$ 7. **Rewrite the equation:** $$(y + 4)^2 = (x - 5)^2 - 9$$ 8. **Take the square root of both sides:** $$y + 4 = \pm \sqrt{(x - 5)^2 - 9}$$ 9. **Solve for $y$:** $$y = -4 \pm \sqrt{(x - 5)^2 - 9}$$ **Final answer:** $$\boxed{y = -4 \pm \sqrt{(x - 5)^2 - 9}}$$ This represents two branches of the solution for $y$ in terms of $x$.