Answer:
[tex]2x^{4}+x^{3}-x+2[/tex]
Step-by-step explanation:
To find which one is prime, let's try to factor them all. We can use the factoring by grouping method.
[tex]x^{3}+3x^{2}-2x-6[/tex]
[tex]x^{3}+3x^{2}[/tex]and[tex]-2x-6[/tex]
[tex]x^{2}(x+3)[/tex]and[tex]-2(x+3)[/tex]
So this one is not prime, since you can still factor it.
[tex]x^{3}+2x^{2}-3x-6[/tex]
[tex]x^{3}+2x^{2}[/tex]and[tex]-3x-6[/tex]
[tex]x^{2}(x-2)[/tex]and[tex]3(x-2)[/tex]
So this one is not prime, since you can still factor it.
[tex]4x^{4}+4x^{2}-2x-2[/tex]
[tex]4x^{4}+4x^{2}[/tex]and[tex]-2x-2[/tex]
[tex]4x^{3}(x+1)[/tex]and[tex]-2(x+1)[/tex]
So this one is not prime, since you can still factor it.
[tex]2x^{4}+x^{3}-x+2[/tex]
[tex]2x^{4}+x^{3}[/tex]and[tex]-x+2[/tex]
[tex]x^{3}(2x+1)[/tex]and -x+2 cannot be further factored.
Therefore, [tex]2x^{4}+x^{3}-x+2[/tex] is a prime.
