Design some resistor with operational amplifier
Here is a circuit that implements:

whatever happens to my input voltage (it can vary depending on certain conditions between 4V DC and 6V DC) I have 5V output




simulate this circuit &ndash; Schematic created using CircuitLab
It involves taking some scissors and disconnecting your input voltage entirely. I suspect that's not what you actually want; that being the case there's a pressing need for you to edit your question.
If you actually want a gain of 2 where your input ranges between 4-6V and your output is centered around 5V, your output would range between 3-7V and you would have (any ratiometric multiple of):



simulate this circuit

Who knows how you want a potentiometer to factor into this. Maybe you don't actually need the original 4-6 pot voltage, and you only need a range of 3-7V, in which case:



simulate this circuit

Or maybe you still want a fixed gain of 2 on an input Ve, and want your pot to represent an adjustable reference to subtract from this voltage:



simulate this circuit
For five different potentiometer positions, this yields

Isn't wild guessing fun?