Final answer:
All provided solutions x = -8, x = -4, and x = 6 satisfy the polynomial equation when substituted back into x^3 + 6x^2 - 40x = 192. Therefore, without additional context, none of the solutions should be eliminated as they are all mathematically valid.
Step-by-step explanation:
The student is asking why some solutions to a polynomial equation do not make sense. For the given equation x^3 + 6x^2 - 40x = 192, the solutions provided are x = -8, x = -4, and x = 6. To determine if these solutions are correct, we substitute them into the equation to see if they satisfy it.
For x = -8: (-8)^3 + 6(-8)^2 - 40(-8) = -512 + 384 + 320 = 192
For x = -4: (-4)^3 + 6(-4)^2 - 40(-4) = -64 + 96 + 160 = 192
For x = 6: (6)^3 + 6(6)^2 - 40(6) = 216 + 216 - 240 = 192
All solutions satisfy the equation, so none of them should be eliminated on the basis of the equation itself. However, without additional context as to the nature of the problem (such as the variables representing something specific in real life where negative values don't make sense), we cannot conclusively eliminate any solution.
To check the answer for reasonableness, re-substitute the solutions into the original equation and verify that the left and right sides are equal, which we have done. Finally, eliminate terms when necessary to simplify the verification process.