Solve a^3+5a^2+6a+8=b^3+5b^2+6b+8

$$a ^ { 3 } + 5 a ^ { 2 } + 6 a + 8 = b ^ { 3 } + 5 b ^ { 2 } + 6 b + 8$$

Solve for a (complex solution)

$$a=\frac{\sqrt{1-10b-3b^{2}}-b-5}{2}$$

$$a=\frac{-\sqrt{1-10b-3b^{2}}-b-5}{2}$$

$$a=b$$

Solve for b (complex solution)

$$b=\frac{\sqrt{1-10a-3a^{2}}-a-5}{2}$$

$$b=\frac{-\sqrt{1-10a-3a^{2}}-a-5}{2}$$

$$b=a$$

Solve for a

$$\left\{\begin{matrix}\\a=b\text{, }&\text{unconditionally}\\a=\frac{\sqrt{1-10b-3b^{2}}-b-5}{2}\text{; }a=\frac{-\sqrt{1-10b-3b^{2}}-b-5}{2}\text{, }&b\geq \frac{-2\sqrt{7}-5}{3}\text{ and }b\leq \frac{2\sqrt{7}-5}{3}\end{matrix}\right.$$

Solve for b

$$\left\{\begin{matrix}\\b=a\text{, }&\text{unconditionally}\\b=\frac{\sqrt{1-10a-3a^{2}}-a-5}{2}\text{; }b=\frac{-\sqrt{1-10a-3a^{2}}-a-5}{2}\text{, }&a\geq \frac{-2\sqrt{7}-5}{3}\text{ and }a\leq \frac{2\sqrt{7}-5}{3}\end{matrix}\right.$$