8x²-2x-1

I assume that you are trying to factor 8x^{2} - 2x - 1

As you probably know, the factors will appear in the form:

(?x ? ?)(?x ? ?)

Where the first and last question marks are numbers and the middle question marks are addition or subtraction signs.

For this problem, I will start with the right side of each factor and work to the left.

Since the constant in the expression is 1, the factors of 1 are 1 and 1. We can replace the last question mark in each factor with a 1

(?x ? 1)(?x ? 1)

The next thing we need to determine is the sign in each parenthesis. Since the sign on the constant is negative, the signs in the parenthesis must be opposites. One sign is positive and one sign is negative.

(?x + 1)(?x - 1)

The hardest part of this problem is determining what is multiplied by x in each of the factors. Since the x^{2} term is multiplied by 8, the numbers must multiply together to equal 8. That means that we will use either 1 and 8 or 2 and 4. We will
need to look at the other terms to decide...

Since the constant has a negative sign in front of it, we are going look for factors that subtract to equal the number multiplied by x. Since this number is -2, we are looking for factors of 8 that subtract to equal 2. These factors are 2 and 4...

The final step is determining which number goes with which x. Since the sign on the 2 is negative, we need to make the larger number (4) become negative when multiplied by the constant. This means we need to put the 4 in the factor that does not have the negative sign.

(4x + 1)(2x - 1)

You can FOIL this to check that the answer is correct.