Is the polynomial $2x^2+3x-2$ prime?

Step 1 ：First, we try to factor the polynomial. A polynomial is prime if it cannot be factored into two or more nontrivial polynomials.

Step 2 ：We can try factoring the polynomial using the AC method. This method involves multiplying the coefficient of $x^2$ (which is 2) by the constant term (which is -2), giving us -4. We then look for two numbers that multiply to -4 and add up to 3 (the coefficient of x). The numbers 4 and -1 meet these criteria.

Step 3 ：We can then rewrite the polynomial as $2x^2+4x-x-2$ and factor by grouping. This gives us $2x(x+2)-1(x+2)$.

Step 4 ：Finally, we can factor out $x+2$, giving us $(2x-1)(x+2)$.