Factoring trinomials is challenging enough, but when the first term has a leading coefficient, it becomes even more difficult. Factoring a trinomial where "a" does not equal 1 requires some guessing and checking, but there are some steps you can take to make the process a little easier before you begin. Be prepared with plenty of paper when you tackle these problems, because it can take several tries to find the right combination of factors for your answer.
Put the problem in standard form, which resembles ax^2 + bx +c. In 14x + 2x^2 + 24, you need to rearrange it by moving the 2x^2 to the front of the equation. This gives you 2x^2 + 14x + 24.
Check if you can factor out any common factors. In the first example, each term is divisible by 2. Factor out the 2 to give you 2 (x^2 + 7x + 12).
Factor the trinomial that is inside the parentheses. Make a list of the factors of 12, then determine which combination would add together to give you 7. The factors of 12 are 1 * 12, 2 * 6, and 3 * 4. The pair that adds together to make 7 is 3 and 4, so the problem factors to 2(x + 3)(x + 4).
Make a list of all the possible factors of the first and last terms for problems that do not contain a common factor, such as 6x^2 + x - 12. The factors of 6 are 1 * 6 and 2 * 3. The factors of -12 are 1 * -12, -1 * 12, 2 * -6, -2 * 6, 3 * -4 and -3 *4.
Choose a pair of factors for each number and create a list of all possible factor combinations. For instance, if you use 2 * 3 and -3 * 4 as the two factors, you could have two possible combinations: (2x - 3)(3x + 4) or (2x + 4)(3x -3).
Use the FOIL method to check if the combinations you have created give you the original trinomial. For these two combinations, (2x - 3)(3x + 4) yields 6x^2 - x - 12, and (2x + 4)(3x - 3) yields 6x^2 - 6x - 12, which are not correct.
Continue trying combinations until you find the correct one. Start by switching the signs in the parenthesis of the two you just tried: (2x + 3)(3x -4) and (2x - 4)(3x +3). The first combination yields 6x^2 + x -12, so it is the correct way to factor this trinomial. You do not need to try the second one because there will be only one correct answer.