To factor the expression 3x² + 2x + 1, we begin by looking for two numbers that multiply to give the product of the coefficient of x² term (which is 3) and the constant term (which is 1). This product is 3 * 1 = 3.
Next, we want these two numbers to add up to the coefficient of the x term, which is 2. However, there are no two numbers that multiply to 3 and also add to 2. Given that this is the case, we can conclude that the expression 3x² + 2x + 1 cannot be factored into simpler binomials with real numbers.
Thus, the expression is already in its simplest form. Alternatively, we could use the quadratic formula to find the roots if needed, but for factoring into binomials with rational coefficients, it remains unreduced.