Another approach, maybe nicer: Using (xy)^2 = 4, we can write the quantity as x + y + (1/4)x^5 + (1/4)y^5 = 4 + (1/4)x^5 + (1/4)y^5. Since x + y = 4 and xy = -2, we have x^2 = 4x + 2. Therefore, x^3 = 4x^2 + 2x = 18x + 8, and x^4 = 18x^2 + 8x = 80x + 36, so x^5 = 80x^2 + 36x = 356x + 160. Because y satisfies the same equation (by symmetry) we have y^5 = 356y + 160. Therefore the desired quantity is equal to 4 + 89x + 40 + 89y + 40 = 84 + 89(x + y) = 84 + 89*4 = 440.
Another approach, maybe nicer: Using (xy)^2 = 4, we can write the quantity as x + y + (1/4)x^5 + (1/4)y^5 = 4 + (1/4)x^5 + (1/4)y^5. Since x + y = 4 and xy = -2, we have x^2 = 4x + 2. Therefore, x^3 = 4x^2 + 2x = 18x + 8, and x^4 = 18x^2 + 8x = 80x + 36, so x^5 = 80x^2 + 36x = 356x + 160. Because y satisfies the same equation (by symmetry) we have y^5 = 356y + 160. Therefore the desired quantity is equal to 4 + 89x + 40 + 89y + 40 = 84 + 89(x + y) = 84 + 89*4 = 440.
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