I think what happened in my thinking was a confusion between the product and sum and its relationship to the constant-so my brain toggled between the product equaling the constant and the product equaling the product of coefficient of x^2 and the constant. Holding a concept steady is still a challenge to me!

Okay, that helps. I find a lot of students have this sort of problem, either losing focus or just forgetting which is sum and which is product.

What I recommend to them is to write down what they are thinking, so they have something to focus on. I tell them, in general, "Think. Then write what you thought. Then think about what you wrote. Then fix it!"

Here, you are factoring 4x^2 + 19x + 12 using the method that requires you to split 19 into the sum of two numbers whose product is 48 (the product of 4 and 12). So we write down:

product = 48

sum = 19.

Then we

**think **-- is that really what I want to do? Either we can go back to the

reasoning behind the method (which unfortunately is not always taught); or just keep some basic fact in mind, such as that the

__product__ of our two numbers is the same as the

__product__ of a and c. That can make it a little more memorable.

Now, having the goal written down, we can carry it out. I don't know how you do your search for numbers, but I either just see a pair that works, or, if that doesn't come to me quickly, start listing pairs of factors of 48:

1*48

2*24

3*16

until I get a pair whose sum is 19, as I just did.

Whether it just came to me or I went through an orderly list, I then check my pair against the criteria I wrote down. Is the sum 19? Yes, 3 + 16 = 19. Is the product 48? Yes, 3*16 = 48. This is especially important when I might get signs wrong, or I might accidentally see a pair of numbers whose sum is 48 and whose product is 19. (For example, if the product were 11 and sum -12.)

Now, the error you specifically mentioned is forgetting whether to make the product c or ac. That is an easy thing to neglect because when a=1, you

*do *just use c. But in that case, c is in fact the same as ac. So what you can do is get used to

**always **using ac, regardless of what a is. So

*whenever *you factor a quadratic trinomial, you can just recite "the product is the product", or whatever helps you.