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Question: Let $ f(x) $ be a polynomial such that
f(x^2 + 2) = x^4 + 4x^2 + 2.
Find $ f(x^2 - 2) $.
Solution: We are given that
Let us denote $ y = x^2 + 2 $, so that $ x^2 = y - 2 $. Then
x^4 = (x^2)^2 = (y - 2)^2 = y^2 - 4y + 4.
Now substitute into the expression for $ f(y) $:
f(y) = x^4 + 4x^2 + 2 = (y^2 - 4y +
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