In order to do this, we need to first consider possible combinations of the coefficients of x and the constant terms. We know that our result will take the following form:
where a and b are factors of the leading coefficient and m and n are factors of the constant term. When FOIL is used to find the resulting trinomial, we get the following expression:
`abx^2 + (bm+an)x + mn`
So, we know the following information based on the trinomial:
1) a and b must be either 1 and 6 or 2 and 3 (assuming we have integer coefficients).
2) m and n are 1 and 12, 2 and 6, or 3 and 4 with one negative value and one positive value (because they need to multiply to -12).
In order to factor this without a calculator, you basically just need to take the shotgun approach to see what combination of numbers will give the following result:
`(bm+an) = -1`
This result is what you're looking for because, if you look at the given trinomial, this is the coefficient of the `x` term.
This might take a few trials for you to find, but you'll eventually see that the following result will come from the possible factors we looked at above:
`(2*4 + 3*(-3)) = -1`
This gives us the following values:
`a = 3, b = 2, m = 4, n = -3`
Notice that you can switch a and b if you also switch m and n. This result then gives you the following factorization:
If you FOIL this out again, you get the original trinomial, confirming we factored this correctly!
I hope this helped!