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Let f = a^4 + 2a^3-2a^2+2a-3

The roots of a^2+2a-3 are -3,1. (or (x+3)(x-1) are factors of this equation)

By remainder theorem (x-a) is a factor of a function f, iff f(a) = 0
We need to show that (x+3) and (x-1) are also factors of a^4 + 2a^3-2a^2+2a-3. This can be shown by the above discussed remainder theorem.

It can be checked that f(1) = 0 and f(-3) = 0, Hence the a^2+2a-3 is a factor of a^4 + 2a^3-2a^2+2a-3.