  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 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 $\GF{2^{4k}}$,
  is re-derived simply from our results.
