It ALL depends on what your connecting out of and into, but to be honest I would be doubtful of the above cable.
What you have there is a Stereo Jack to two Mono Jacks. if you notice on the stereo (single sided) end there are two wee black lines that split the metal into three sections. These are called the Tip, the Ring and the Sleeve and it is for connecting to a stereo (or balanced but that’s a whole different matter) source, so the Tip will carry the left channel, the Ring will carry the right channel and the sleeve will carry the common ground. This is then split to two mono jacks with just a Tip and Sleeve, so the white jack will have the left and a connection to the ground, and the red jack will have the right signal and a connection to ground.
In short, your taking a stereo signal with Left and right and splitting it into two separate connections, one left and one right.
First problem will be the connection to the amps. You mention it has (as well as the XLR) two connections, a red and a white one and thats the ones you are planning to use. The problem is that almost always, these will be phono connections (like what goes into the back of a mixer) and now while the cable you have above has a red and white marker on it these are jacks. It may be that the amp does indeed have jack inputs but i would be very doubtful and want to double check first.
now on the end that is attaching to your laptop/sound card, as said its very hard to narrow it down without knowing what your using but its more than likely an earbud/headphone type socket if its coming straight from your laptop. What you have pictured above is somewhat correct in that it is a stereo jack your wanting to split into two mono connections, but what you have pictured is the larger version, 1/4 inch jack that would be on the end of say DJ headphones. but if its coming straight from your sound card then it’ll almost certianly be the smaller type of jack that you would get on headphones for your ipod or whatever, 3.5mm (or 1/8 inch) jack.
lol, so Ok after all of that what i would GUESS you are looking for is something like this…

where you have the smaller, but still stereo, jack connected to two phono connections.
NOTE: sometimes jacks are called PHONE connectors because in the old days when the telephone companies needed wee women to sit and plug in wires to make the connections thats what they had on the end of them, and this can sometimes get confused with PHONO with an o on the end, which is the type of little connections that would be say on the back of a DJ mixer.
NOTE 2: you mention you cant use XLR because you dont have a mixer but it is defo possible to get a cable that has stereo jack to two XLR’s or phono’s to two XLR’s.
but again yeah this is all guesswork if you can post the type of soundcard you are using and maybe even the brand or model of amp your plugging into i can tell you exactly what you’ll need.
k