Solve (y+k)^2+y^2+2(y+k)-4y-1=0

$$( y + k ) ^ { 2 } + y ^ { 2 } + 2 ( y + k ) - 4 y - 1 = 0$$

Solve for k (complex solution)

$$k=\sqrt{2+4y-y^{2}}-y-1$$

$$k=-\left(\sqrt{2+4y-y^{2}}+y+1\right)$$

Solve for y (complex solution)

$$y=\frac{\sqrt{3-6k-k^{2}}-k+1}{2}$$

$$y=\frac{-\sqrt{3-6k-k^{2}}-k+1}{2}$$

Solve for k

$$k=-\left(\sqrt{2+4y-y^{2}}+y+1\right)$$

$$k=\sqrt{2+4y-y^{2}}-y-1\text{, }y\geq 2-\sqrt{6}\text{ and }y\leq \sqrt{6}+2$$

Solve for y

$$y=\frac{\sqrt{3-6k-k^{2}}-k+1}{2}$$

$$y=\frac{-\sqrt{3-6k-k^{2}}-k+1}{2}\text{, }k\geq -2\sqrt{3}-3\text{ and }k\leq 2\sqrt{3}-3$$