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Let $F(X)=X^{2^{2k}+2^k+1}$ be the power function over the finite field$\GF{2^{4k}}$ which is known as the Bracken-Leander function. In\cite{BCC10,BL10,CV20,Fu22,XY17}, it was proved that the number of solutions in$\GF{q^4}$ to the equation $F(X)+F(X+1)=b$ is in $\{0,2,4\}$ for any $b\in\GF{q^4}$ and the number of the $b$ giving $i$ solutions have been determinedfor every $i$. However, no paper provided a direct and complete method to solvesuch an equation, and  this problem remained open. This article presents a direct technique toderive an explicit solution to that equation. The main result  in \cite{BCC10,BL10,Fu22,XY17}, determining differential spectrum of  $F(X)=X^{2^{2k}+2^k+1}$ over $\GF{2^{4k}}$,  is re-derived simply from our results.

Solving $X^{2^{2k}+2^{k}+1}+(X+1)^{2^{2k}+2^{k}+1}=b$ over $\GF{2^{4k}}$