Solve in $\ \mathbb{Z^2}$:
$$8a^2(a+b)=4ab(a+b)+b^3$$
2026-05-03 18:27:30.1777832850
Solving $8a^2(a+b)=4ab(a+b)+b^3$ over the integers
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true if both $a,b=0.$ Otherwise, take $a \neq 0$ and $x = \frac{b}{a}.$ Your equation asks for a rational root of $$ x^3 + 4 x^2 - 4 x - 8 $$ but there are none.