An algebra problem by ابراهيم فقرا

We note that $\begin{aligned} (a + b)(a^2 + b^2) & = \; a^3 + b^3 + ab(a + b) \\ (6 - 2c)(2 - \tfrac12c^2) & = \; -\tfrac12c^3 + \tfrac12(6 - 2c)\big[(6 - 2c)^2 - 2 + \tfrac12c^2\big] \end{aligned}$ This is a cubic equation for $c$ with roots $2$ , $\tfrac{1}{8}\big(19 \pm i\sqrt{119})$ . If $c=2$ we find that $a,b = 1 \pm i$ , and $k=\boxed{24}$ . The other two values of $c$ yield complex values for $k$ , so can be ignored.