This is the problem with printers and monitors - they never give you the same results
Does the image or document you are printing have it's own colour profile? If not then if it's possible assign one to it to help the printer translate the colours correctly. If you've already tried this or if this does not work it may be that the bright red is simply out of the printers gamut range.
I've encountered this myself before and through trial-and-error adjustments I discovered that if I over-saturate the colours in the image/document as they appear on screen then the printed results end up more akin to what I was expecting. However, this is not a guaranteed result as some colours (depending on printer, software and so on) do not play well
Red is always a difficult colour to reproduce successfully.