0
4x squared - x cubed = 506
x squared times (4 - x) = 506
prime factors of 506 are 2, 11 & 23 so possible factors are 22 x 23 and 11 x 46.
None of these are square numbers so first impression is that there is no solution in integers. I also suspect that any solution will be negative.
More later!
Later: Nearest I can get without going to ten-thousandths is x = -6.834.
4xsquared = 186.814, x cubed = -319.172 which sum to 505.986.
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