jmc

The graph of $f(x)=x^2+bx+c$ is an upward-facing parabola, and the condition $$f(2+t)=f(2-t)$$tells us that the axis of symmetry of the parabola is the line $x=2.$ Thus, $f(x)$ is an increasing function of $|x-2|.$ In other words, the farther $x$ is from 2, the greater $f(x)$ is.



Hence, $\boxed{f(2)<f(1)<f(4)}.$