5. The quadratic function $f(x)=a+b x-x^{2}$ satisfies $f(1+x)=f(1-x)$ for any real number $x$, and $f(x+m)$ is an increasing function on $(-\infty, 4]$. Then the range of the real number $m$ is $\qquad$
5. $m \in(-\infty,-3]$

From $f(1+x)=f(1-x)$, we know that $x=1$ is the axis of symmetry of $f(x)$, thus $m+4 \leqslant 1, m \leqslant-3$.