about the complex number

Question

Let $a+ ib$ be the complex root of $f(x)=x^3+2x+1$. I want to find $a$.

My Try: $f(a+ib)=0$. It follow that $$(a^3-3ab^2+2a+1)=(-b^3+3a^2b+2b)=0$$

Answer 1

I hope it means $a$ and $b$ are reals.

If so, $x^3+2x+1$ is divisible by $$(x-a-bi)(x-a+bi)=x^2-2ax+a^2+b^2,$$ which is possible for the following factoring only. $$x^3+2x+1=(x+2a)(x^2-2ax+a^2+b^2),$$ which says that $a=-\frac{1}{2}x_1$, where $x_1$ is a real root of the polynomial $x^3+2x+1$.

Now, use the Cardano's formula: $$x_1=\sqrt[3]{-\frac{1}{2}+\sqrt{\frac{1}{4}+\frac{8}{27}}}+\sqrt[3]{-\frac{1}{2}-\sqrt{\frac{1}{4}+\frac{8}{27}}}=...$$