Out of the four polynomials you listed, the prime polynomial is: 2x^4 + x^3 - x + 2.
A prime polynomial is one that cannot be factored into any non-trivial products of polynomials with coefficients in the same field as the original polynomial.
We usually work with polynomials with integer coefficients, so in this case, we're looking for a polynomial that cannot be factored into products of polynomials with integer coefficients.
Here's why the others aren't prime:
x³ + 3x²-2x-6: This factors as (x+2)(x-3), making it composite (not prime).
x²-2x²+3x-6: This can be rearranged as -(x²-2x) + 3x-6, which factors into -x(x-2) + 3(x-2), making it composite.
4x^4 + 4x³-2x-2: This can be factored as 2(2x^2 + 2x - 1) further factoring the second term as 2(x-1)(x+1), making it composite.
Therefore, the only polynomial that doesn't factor further into simpler polynomials is 2x^4 + x^3 - x + 2, making it the prime polynomial.