Subject 2. Solve in integers the equation: $2^{x+1}=x^{2}+x+1$.

Mathematical Gazette 1987
Subject 2. $x \leq-2 \Rightarrow 2^{x+1} \leq \frac{1}{2}x^{2}+x+1$ (1p); verification $x=1 \mathbf{( 1 p}$ ); correct proof of the induction step (2p).