Dr. Taylor, I have been racking my brain for 30 minutes on this problem, and I don't see any possible answer other than the one I have given. I used the projection formula we learned in class and even went through the problem with a fellow classmate who agreed that my work seemed correct. Thank you for your time.
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well, you have the vector projection correct. The scalar projection is b.a^ = b.a/||a|| which is
[(-1)x3 + 4x(-3) + (-1)x(-4)]/√(3^2 + (-3)^2 + (-4)^2) = [-11]/√34
So, your issue is you're using b.a/||a||^2 (and maybe also that you didn't look at the textbook?)
