The polynomial p(x) = 2x3+10x2+10x+2 clearly has a root at x = -1, leading to the factorization

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The roots of the quadratic factor can be found by completing the square or with the quadratic formula.  They turn out to be

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Now we know that -1 = -tan(45).  We can use the half-angle formula for tangent to see that the remaining roots are -tan(15) and -tan(75). The half-angle formula says

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Taking A = 30 leads to

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and with A = 150 we find

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This shows that the quadratic factor of the polynomial has -tan(15) and -tan(75) as roots.