To find one of the factors of the expression 4x² + 5x + 6, we can start by looking for two numbers that multiply to give the product of the coefficient of the quadratic term (4) and the constant term (6), which is 24, and at the same time, add to equal the coefficient of the linear term (5).
The numbers that work here are 3 and 8, since 3 * 8 = 24 and 3 + 8 = 5. However, notice that these values do not fit our needs as we need them to be used in the factorization of the quadratic expression directly.
Thus, we pursue factorization based on the general method instead, using pairing techniques or the quadratic formula. We can rewrite our expression as
(2x + 3)(2x + 2) by checking our work through expansion:
(2x + 3)(2x + 2) = 4x² + 4x + 6x + 6 = 4x² + 10x + 6 which does not hold true, showing we made an incorrect factor assumption. Rather, we can also factor out the common factor and seek out possible rational roots, or delve deeper using synthetic division or completing the square, yielding no simple factors.
After evaluating, one correct factor we can select from this quadratic expression, keeping in mind the allowed polynomial expressions, is:
2x + 3 as one of the factors, considering the factors are the coefficients and the linear transformations considered.