The expression (x^2 - 2x - 4) factors as (x + 1 - sqrt(5))*(x + 1 + sqrt(5)), where sqrt(5) is the square root of 5. This quadratic polynomial is factored by completing the square, a method in which a particular number is both added to and subtracted from the polynomial in order to factor the polynomial.
To complete the square, begin by writing the polynomial as x^2 - 2x = 4. Note that the left-hand side can be factored into (x - 1)^2 if 1 is added to the expression. The number 1 must also be added to the right-hand side so the polynomial remains the same: x^2 - 2x + 1 = 4 + 1. Now the polynomial simplifies to (x - 1)^2 = 5. Solve this expression for the value of x by taking the square root of both sides and adding 1 to both sides. The right-hand side can be either the positive or the negative square root of 5; thus, there are two possible solution values for x: x = 1 + sqrt(5) or x = 1 - sqrt(5). A quadratic polynomial that factors can always be written as (x + A)*(x + B), where A and B are the two values of x at which the polynomial is equal to zero. In this case, the polynomial x^2 - 2x - 4 factors as (x + 1 + sqrt(5))*(x + 1 - sqrt(5)).