$$4x^{2}+2x+1=0$$

$$x=\frac{-2±\sqrt{2^{2}-4\times 4}}{2\times 4}$$

$$x=\frac{-2±\sqrt{4-4\times 4}}{2\times 4}$$

$$x=\frac{-2±\sqrt{4-16}}{2\times 4}$$

$$x=\frac{-2±\sqrt{-12}}{2\times 4}$$

$$x=\frac{-2±2\sqrt{3}i}{2\times 4}$$

$$x=\frac{-2±2\sqrt{3}i}{8}$$

$$x=\frac{-2+2\sqrt{3}i}{8}$$

$$x=\frac{-1+\sqrt{3}i}{4}$$

$$x=\frac{-2\sqrt{3}i-2}{8}$$

$$x=\frac{-\sqrt{3}i-1}{4}$$

$$x=\frac{-1+\sqrt{3}i}{4}$$ $$x=\frac{-\sqrt{3}i-1}{4}$$

$$4x^{2}+2x+1=0$$

$$4x^{2}+2x+1-1=-1$$

$$4x^{2}+2x=-1$$

$$\frac{4x^{2}+2x}{4}=-\frac{1}{4}$$

$$x^{2}+\frac{2}{4}x=-\frac{1}{4}$$

$$x^{2}+\frac{1}{2}x=-\frac{1}{4}$$

$$x^{2}+\frac{1}{2}x+\left(\frac{1}{4}\right)^{2}=-\frac{1}{4}+\left(\frac{1}{4}\right)^{2}$$

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=-\frac{1}{4}+\frac{1}{16}$$

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=-\frac{3}{16}$$

$$\left(x+\frac{1}{4}\right)^{2}=-\frac{3}{16}$$

$$\sqrt{\left(x+\frac{1}{4}\right)^{2}}=\sqrt{-\frac{3}{16}}$$

$$x+\frac{1}{4}=\frac{\sqrt{3}i}{4}$$ $$x+\frac{1}{4}=-\frac{\sqrt{3}i}{4}$$

$$x=\frac{-1+\sqrt{3}i}{4}$$ $$x=\frac{-\sqrt{3}i-1}{4}$$

$$x ^ 2 +\frac{1}{2}x +\frac{1}{4} = 0$$

$$r + s = -\frac{1}{2} $$ $$ rs = \frac{1}{4}$$

$$r = -\frac{1}{4} - u$$ $$s = -\frac{1}{4} + u$$

$$(-\frac{1}{4} - u) (-\frac{1}{4} + u) = \frac{1}{4}$$

$$\frac{1}{16} - u^2 = \frac{1}{4}$$

$$-u^2 = \frac{1}{4}-\frac{1}{16} = \frac{3}{16}$$

$$u^2 = -\frac{3}{16}$$ $$u = \pm\sqrt{-\frac{3}{16}} = \pm \frac{\sqrt{3}}{4}i $$

$$r =-\frac{1}{4} - \frac{\sqrt{3}}{4}i = -0.250 - 0.433i$$ $$s = -\frac{1}{4} + \frac{\sqrt{3}}{4}i = -0.250 + 0.433i$$