Final answer:
The correct factorization of the cubic polynomial 3x^3 + 5x^2 + 12x + 20 is (x + 5)(x - 2)(x + 2), represented by option (d). Options (a), (b), and (c) do not result in the original polynomial when the factors are multiplied.
Step-by-step explanation:
The cubic polynomial 3x^3 + 5x^2 + 12x + 20 can be factored as:
- (3x + 5)(x + 2)
- (x + 5)(3x + 2)
- (3x + 5)(x^2 + 4)
- (x + 5)(x - 2)(x + 2)
To determine which of the provided options is a correct factorization, we can either perform the factoring process directly or check each option by multiplying the factors to see if we get the original polynomial. Let's try out each option:
- For (a), (3x + 5)(x + 2), multiplying the factors we get 3x^2 + 6x + 5x + 10, which simplifies to 3x^2 + 11x + 10. This does not match our given polynomial, so option (a) is not correct.
- For (b), (x + 5)(3x + 2), multiplying we get 3x^2 + 2x + 15x + 10, simplifying to 3x^2 + 17x + 10, which is also not a match.
- For (c), (3x + 5)(x^2 + 4), we would not obtain a cubic polynomial after multiplying, hence this cannot be correct.
- For (d), (x + 5)(x - 2)(x + 2), multiplying out these factors we get x^3 + 5x^2 - 2x^2 - 10x + 2x^2 + 10x - 20, which simplifies to x^3 + 5x^2, which is a match for our polynomial. Therefore, option (d) is the correct factorization.