 So the reason that we've been focusing on factoring monic polynomials is that if it factors at all, the factors of a monic polynomial must be of the form x plus a. But a non-monic polynomial can have factors of other forms. So if I multiply 2x plus 3 by x plus 4, I get. Or if I multiply 3x plus 4 by 2x minus 1, I get. And so here we see that if I have a non-monic polynomial, it could be that one of the factors is of the form x plus a. Or it could be that none of the factors is of this form. And this makes factoring even the simplest non-monic polynomial extremely difficult. Even in the easy case, there's still a lot of work ahead of us. So remember to factor x squared plus px plus q, we only need to list all possible factorizations of q. If I want to factor ax squared plus bx plus c, we need to keep track of factorizations of both ax squared and c. And remember, trial and error factoring requires trying every possible factor. Now, this could be a lot of bookkeeping. And that's OK. Arithmetic is bookkeeping. An algebra is generalized arithmetic, so you should expect bookkeeping. And we have all sorts of bookkeeping tricks to keep everything organized. And so in order to keep track of which factors we've tried, we'll organize them into a table. And so maybe we'll set our columns as the factorizations of ax squared and our rows as the factorizations of c. So let's try to factor this non-monic polynomial. So remember, we should always remove common factors first. And we see that the three terms of this polynomial have no common factors. So that doesn't really help us. This is actually a non-monic polynomial. So let's organize our possibilities. So we want to find our factorizations of 2x squared, the quadratic term. Well, that's easy. The only factorization there is x and 2x. We also want to find our factorizations of minus 5. And so our factorizations of minus 5 are, since the only factorization of 2x squared is x and 2x, we know that if this expression factors, it'll factor as x something, 2x something. So I'll try our first pair of factors 1 and negative 5. Is this x plus 1 times 2x minus 5? Now, before we dismiss this as a possibility, notice that we can actually switch the places of our constant terms. And this will actually give us a different factorization. And what that means is that for every possible pair of factors of c, we have to check two possibilities. And so what we'll do is we'll actually split this column to remind us that we actually have to check both possibilities. So is this going to factor as x minus 5 times 2x plus 1? No. How about using the factorization of the constant minus 1 and 5? So we check out x minus 1 times 2x plus 5. But remember, when factoring a non-monic polynomial, remember you can change where you place the factors of the constant term. So we'll change this to x plus 5 times 2x minus 1 and check. And we have our factorization. And now for the bad news. That was about the simplest possible case where you have to factor a non-monic polynomial. Factoring requires patience and persistence. There is no substitute. So let's try this problem. And so we'll start by finding the factorizations of 6x squared, which are and the factorizations of minus 7r, organizing our possibilities. And remember, when factoring a non-monic polynomial, you can change where you place the factors of the constant term. If we do each of the possibilities for the factorization of the constant term, we actually have two things to check. So we'll try x and 6x first with 1 and negative 7. So does this factor as x plus 1 times 6x minus 7? Let's switch to the location of the constant terms. Well, let's try negative 1 and 7. So does this factor as x minus 1 times 6x plus 7? Nope. Switch to the location of the constant terms. Nope. So we have to move on to the next possibilities for 6x squared. That's 2x and 3x. So again, we'll try our first factorization for the constants, 2x plus 1 and 3x minus 7. Nope. Wait, no, that works. And so we get our factorization. So it's important to remember, trial and error factoring requires trying every possible factor. Patience and persistence is absolutely essential and there is no substitute.