Let's analyze the given polynomials one by one to determine which one is prime.
A polynomial is considered prime if it cannot be factored into smaller-degree polynomials with integer coefficients.
1. Polynomial: [tex]\( 3x^3 + 3x^2 - 2x - 2 \)[/tex]
2. Polynomial: [tex]\( 3x^3 - 2x^2 + 3x - 4 \)[/tex]
3. Polynomial: [tex]\( 4x^3 + 2x^2 + 6x + 3 \)[/tex]
4. Polynomial: [tex]\( 4x^3 + 4x^2 - 3x - 3 \)[/tex]
After thorough analysis, the prime polynomial among the four given polynomials is the first one:
[tex]\[ 3x^3 + 3x^2 - 2x - 2 \][/tex]
This polynomial cannot be factored into smaller-degree polynomials with integer coefficients. Therefore, the correct answer is the first polynomial.
