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I'm clueless as to why as your integrand is (F / 6), you totally lost me there. But that you did it wrong should be obvious because you can check yourself by doing your integral with the force, that you determined to be -1.5*x, and integrating it from 6 to 0. When you do this you should get the original work required to compress the spring 6 inches (54 inch/lb), but you don't, you come up with 27 inch/lb instead. So something is clearly wrong already.
Try it this way:
W = [integral from 6 to 0] F(x) dx
substituting in for F(x) by F(x) = - k * x
W = [integral from 6 to 0] (- k * x) dx
Now set W = 54 inch/lb and solve the integral and then you can determine k. Then set up your new integral from 12 to 0 with your determined value for k and that will give the work required to compress the spring 12 inches.
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