Non-inverting amplifier with negative supply rail
Remove R3 and it'll work as you require.

Edit

When input is 0V, the output is -1.25V. Perhaps missing the minus sign off was just a typo. Also assuming Vref is still +5V and that's a dash not a minus sign.

That is a non-inverting amplifier, the input voltage goes into the non-inverting input. If the input is +ve with respect to Vref then the output is +ve with respect to Vref. If the input is -ve with respect to Vref then the output is -ve with respect to Vref. It's a more rarely seen non-inverting amplifier because Vref isn't at 0V.

You'd probably have trouble finding a 4k resistor. I'd recommend R1=12k and R2=3k to get your 4X ratio. (E24 series).

Equation is:-

Vout = (Vin - Vref)(R2/R1) + Vin

where in your case Vref =+5V

You seem to perhaps be designing a variable voltage power supply, are you using a PIC to control it?