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Here is the problem: 2x to the power 4 plus 2x cubed plus 2x squared plus 4x plus 3 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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No further rational roots; approximate the remaining 4 roots numerically. The Durand–Kerner method iterates on all roots at once until they stop changing.

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Collect the real solutions.

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So the answer is No real solutions; complex: 0.41383593 plus or minus 1.168397i, negative 0.91383593 plus or minus 0.37577205i.

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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.