Determine parameter so that the absolute value of real solution of the equation is larger than the modulo of complex solution

Question

Given the equation: $$x^3+x+\lambda=0$$ determine real parameter $\lambda$ so that the real solution is greater by absolute value than modulo of the complex solutions. My attempt: Let $x_1$ be the real root and $x_2 = a+ib$ one of the complex roots. Then the third root is $x_3=a-ib$. From here, by using Vieta's formulas we have: $$x_1 + a+ib+a-ib =x_1+2a= 0$$ $$x_1(a+ib) + x_1(a-ib) + a^2 +b^2 = 2ax_1 + a^2 + b^2 = 1$$ $$x_1(a^2+b^2) = -\lambda$$ but i seem to be stuck here.

Answer 1

$$x_1=-2a\quad=>\quad b^2=3a^2+1\quad=>\quad\lambda=2a(4a^2+1).$$