Op-amp offset and multiplication
If you are dealing with a single ended input signal you could do it with a fully differential op amp. The simpler ones have a fully diff output, i.e. the output is not referenced to ground but is the difference between two terminals. You build the usual non inverting amp with the inputs and the positive output, and feed \$-4.25\$V to the negative output. This might work but not all op amps like an output to be set to a specific voltage.

If you can use an instrumentation amplifier that would be much better. It is not an opamp but it is a single component. INAMPS need no external feedback, that's built in, and they usually have some very precise preset gain, or you can set it eithger with a control voltage or digitally. The output usually have a reference terminal: that is where you want to hook your \$-4.25\$V, and that terminal is though exactly for this purpose.

To vary the offset you can set the ref voltage with a potentiometer, maybe add some safety resistors in order to not let the ref terminal swing the full \$\pm 15\$V: the inamp might not like it as well as your subsequent circuit.

As hinted in the comments, if your supplies are not well regulated using a mere divider to get the \$-4.25\$V won't work: you will need a linear regulator and a divider after that.

If you absolutely need to do that with a single ended, single op amp there is a circuit:







simulate this circuit &ndash; Schematic created using CircuitLab

Some analysis leads to:
$$
V_{out} = \left(1+\frac{R_2}{R_1}\right)\cdot\frac{1}{R_3+R_4}\left(V_sR_4+ER_3\right) 
$$
that is in the form \$V_{out}=\alpha V_s + \beta E\$ with \$\alpha\$ and \$\beta\$ positive. 

Just work out some values for the resistors if you want to build this last circuit, but the INAMP solution is the best you can get.

pros and cons of each solution:

INAMP pros: it's designed to do what you want, your only concern is the offset voltage but that is a concern for every solution. cons: costs and power consumption.

FULLY DIFF AMP pros: it's easy and cheap to set up, costs and power are just like a single ended opamp. cons: it probably won't work since you try to set one of the output rails to a fixed voltage

SINGLE OPAMP pros: dirty cheap and easy, fits the spec. cons: do you see that ton of resistors? they are bad. they are not precise, they need space, they are bad creatures. Plus \$\alpha\$ and \$\beta\$ are somewhat connected, i.e. you can't vary one of them leaving the other untouched if you act on a single component.