determine an equation in simplified form for the family of quartic functions with zeros -1-√5 and -1+√5 and 2+√2 and 2-√2

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determine an equation in simplified form for the family of quartic functions with zeros -1-√5 and -1+√5 and 2+√2 and 2-√2

Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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We first find a quadratic equation with \[-1-\sqrt{5}\] and \[-1+\sqrt{5}\] as roots. We find the sum and product of the roots. Sum of roots: - 2 Product of roots: - 4 Now, we reverse the sign of the sum of the roots. That is 2. The equation is \[x^2 +2x-4=0\]. For the second quadratic equation. The equation must have \[2+\sqrt2 \] and \[2-\sqrt2\] as roots. Sum of roots: 4 Product of roots: 2 Again, invert the sign of the sum of roots. That is - 4. The equation is \[x^2 -4x+2=0\]. Multiply: \[(x^2 + 2x - 4)(x^2 -4x + 2) = 0\]. Then, the result is your final answer. Hope this helps a lot :)

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