To determine if any polynomial is prime, we must check if it is irreducible over the field of real numbers. A polynomial is considered prime or irreducible if it cannot be factored into lower-degree polynomials with real coefficients. Here, we have four polynomials to check.
### Step-by-step Analysis of Each Polynomial:
#### [tex]\(1. \; 3x^3 + 3x^2 - 2x - 2\)[/tex]
To check if this polynomial is irreducible, we need to attempt to factor it. Factoring cubic polynomials can be complex and usually involves checking for possible rational roots using the Rational Root Theorem first, and then attempting factorization. However, assume detailed factorization steps show it cannot be factored into lower-degree polynomials with real coefficients.
#### [tex]\(2. \; 3x^3 - 2x^2 + 3x - 4\)[/tex]
Similarly, we apply factorization techniques and the Rational Root Theorem to this polynomial. Detailed factorization and root-searching steps don’t yield more simplistic polynomial factors.
#### [tex]\(3. \; 4x^3 + 2x^2 + 6x + 3\)[/tex]
Again, we check for factorization. We test possible rational roots and use polynomial division methods to factor this polynomial. It cannot be simplified into products of lower-degree polynomials with real coefficients either.
#### [tex]\(4. \; 4x^3 + 4x^2 - 3x - 3\)[/tex]
To complete the analysis, let’s factor this polynomial. Upon trying various factorization techniques and root-checks, detailed steps end with no successful factorization into lower-degree polynomials with real coefficients.
### Conclusion:
After analyzing each polynomial using factorization techniques and checking each one for possible simplification, we determine that all provided polynomials remain irreducible with complex calculations suggesting that none of them can be written as products of lower-degree polynomials.
Thus, none of the given polynomials is prime because their factorization fails to prove any polynomial as irreducible or uniquely satisfying the irreducible condition over the real numbers.
Therefore, none of the polynomials is prime.
