Equal sums of two cubes of binary quadratic forms

We give a complete description of all solutions to the equation $f_1^3 + f_2^3 = f_3^3 + f_4^3$forquadraticforms$f_j \in \mathbb C[x,y]$ and show how Ramanujan’s example can be extended to three equal sums of pairs of cubes. We also give a complete census in counting the number of ways a sextic $p \in \mathbb C[x,y]$canbewrittenasasumoftwocubes\mathrm{.}Theextremeexampleis$p(x,y) = xy(x^4-y^4)$, which has six such representations.