Which polynomial is prime? \( x^{3}+3 x^{2}-2 x-6 \) \( x^{3}-2 x^{2}+3 x-6 \) \( 4 x^{4}+4 x^{3}-2 x-2 \) \( 2 x^{4}+x^{3}-x+2 \)

To determine which polynomial is prime, we need to check if the polynomial can be factored into simpler polynomials. If it cannot be factored, then it is considered a prime polynomial. Let's analyze each polynomial given: 1. \( x^{3}+3x^{2}-2x-6 \) 2. \( x^{3}-2x^{2}+3x-6 \) 3. \( 4x^{4}+4x^{3}-2x-2 \) 4. \( 2x^{4}+x^{3}-x+2 \) We will factor each polynomial to determine if it is prime. Factor the expression by following steps: - step0: Factor: \(x^{3}+3x^{2}-2x-6\) - step1: Rewrite the expression: \(x^{3}-2x+3x^{2}-6\) - step2: Factor the expression: \(x\left(x^{2}-2\right)+3\left(x^{2}-2\right)\) - step3: Factor the expression: \(\left(x+3\right)\left(x^{2}-2\right)\) The polynomial \( x^{3}+3x^{2}-2x-6 \) can be factored as \( (x+3)(x^{2}-2) \). Since it can be factored, it is not a prime polynomial. Now, let's factor the second polynomial: 2. \( x^{3}-2x^{2}+3x-6 \) Factor the expression by following steps: - step0: Factor: \(x^{3}-2x^{2}+3x-6\) - step1: Rewrite the expression: \(x^{3}+3x-2x^{2}-6\) - step2: Factor the expression: \(x\left(x^{2}+3\right)-2\left(x^{2}+3\right)\) - step3: Factor the expression: \(\left(x-2\right)\left(x^{2}+3\right)\) The polynomial \( x^{3}-2x^{2}+3x-6 \) can be factored as \( (x-2)(x^{2}+3) \). Since it can be factored, it is not a prime polynomial. Next, let's factor the third polynomial: 3. \( 4x^{4}+4x^{3}-2x-2 \) Factor the expression by following steps: - step0: Factor: \(4x^{4}+4x^{3}-2x-2\) - step1: Factor the expression: \(2\left(2x^{4}-x+2x^{3}-1\right)\) - step2: Factor the expression: \(2\left(x\left(2x^{3}-1\right)+2x^{3}-1\right)\) - step3: Factor the expression: \(2\left(x+1\right)\left(2x^{3}-1\right)\) Unfortunately, the fourth polynomial \( 2x^{4}+x^{3}-x+2 \) cannot be factored using the available methods. Therefore, we cannot determine if it is a prime polynomial based on the given information.