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Here is the problem: x to the power 4 minus 2x cubed minus x plus 2 equals 0.

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A polynomial equation of degree 4 in x.

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Start with the equation.

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Move every term to one side.

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Test candidates from the rational root theorem: x equals 1 is a root. Synthetic division by (x minus 1) leaves a polynomial of degree 3.

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Test candidates from the rational root theorem: x equals 2 is a root. Synthetic division by (x minus 2) leaves a polynomial of degree 2.

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Solve the remaining quadratic.

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Compute the discriminant.

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delta less than 0, so there are no real roots; the two roots are complex conjugates.

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Apply the quadratic formula with the square root of negative 1 equals i.

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Simplify.

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Collect the real solutions.

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Now look at the graph of the left side minus the right side.

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It crosses zero exactly at the solutions we found.

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So the answer is x equals 1, x equals 2; complex: negative 0.5 plus or minus 0.8660254i.

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A degree-4 polynomial has exactly 4 roots counted with multiplicity (some may be complex). Exact rational roots are found first, then the rest are solved exactly or approximated.

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x equals 1: left side equals 0, right side equals 0, which checks out.

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x equals 2: left side equals 0, right side equals 0, which checks out.