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

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