To determine which product of prime polynomials is equivalent to the given polynomial [tex]\(4x^5 - 6x^4 + 4x^3 - 6x^2\)[/tex], let's go through a step-by-step factorization process.
1. Extract Common Factor:
First, notice that each term in the polynomial [tex]\(4x^5 - 6x^4 + 4x^3 - 6x^2\)[/tex] has a common factor of [tex]\(2x^2\)[/tex]. We can factor this out:
[tex]\[
4x^5 - 6x^4 + 4x^3 - 6x^2 = 2x^2(2x^3 - 3x^2 + 2x - 3)
\][/tex]
2. Factorize the Polynomial Further:
Next, we need to factor the cubic polynomial [tex]\(2x^3 - 3x^2 + 2x - 3\)[/tex]. This may involve recognizing patterns or using polynomial factorization techniques.
3. Identify Prime Factors:
Further factorizing [tex]\(2x^3 - 3x^2 + 2x - 3\)[/tex], we are left with:
[tex]\[
2x^3 - 3x^2 + 2x - 3 = (2x-3)(x^2 + 1)
\][/tex]
4. Combine All Factors:
Finally, combining all the factors together, we get:
[tex]\[
4x^5 - 6x^4 + 4x^3 - 6x^2 = 2x^2 \cdot (2x - 3) \cdot (x^2 + 1)
\][/tex]
This matches with one of the given options. Therefore, the product of prime polynomials equivalent to [tex]\(4x^5 - 6x^4 + 4x^3 - 6x^2\)[/tex] is:
[tex]\[ \boxed{2x^2(x^2 + 1)(2x - 3)} \][/tex]
