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 \((2x - 1)(x + 2)\).