4.1. Find the sum of all roots of the equation $x^{2}-31 x+220=2^{x}\left(31-2 x-2^{x}\right)$.
Answer: 7. Solution. The original equation is equivalent to the equation $\left(x+2^{x}\right)^{2}-31\left(x+2^{x}\right)+220=0 \Leftrightarrow\left[\begin{array}{l}x+2^{x}=11, \\ x+2^{x}=20 .\end{array}\right.$

Each of the equations in this system has no more than one root, as the function $f(x)=x+2^{x}$ is monotonically increasing. The first equation has a root $x=3$, and the second has a root $x=4$. The sum of the roots is 7.