Completing the square rewrites a quadratic so the variable appears only once, inside a squared term. On the SAT it is the technique behind vertex questions, behind converting a circle equation to standard form, and behind any question asking which equivalent form shows a minimum or maximum most clearly.
The rule
Take half the coefficient of x, square it, and add and subtract that number so the expression becomes a perfect square plus a constant.
The goal is to turn x squared plus bx into a perfect square. Take half of b, square it, and add that number, then subtract it again so nothing changes. For x squared plus 8x plus 5, half of 8 is 4 and 4 squared is 16. Write x squared plus 8x plus 16 minus 16 plus 5, which groups into the quantity x plus 4 squared minus 11.
That new form is doing something useful. Since a squared term can never be negative, the smallest value of the quantity x plus 4 squared is zero, which happens when x equals negative 4. So the smallest value of the whole expression is negative 11, occurring at x equal to negative 4. The vertex is at the point with coordinates negative 4 and negative 11, read straight off the rewritten form.
Notice the sign flip on the x-coordinate. In the quantity x plus 4 squared, the vertex sits at negative 4, not positive 4. In the quantity x minus 6 squared, the vertex sits at positive 6. The number inside the parentheses always appears with the opposite sign in the vertex, and the SAT includes both signs among the answer choices.
With a leading coefficient, factor it out of the first two terms before you complete anything. For 2x squared plus 12x plus 1, write 2 times the quantity x squared plus 6x, plus 1. Half of 6 is 3 and 3 squared is 9, so inside the parentheses you build x squared plus 6x plus 9, which is the quantity x plus 3 squared. Because that 9 sits inside a factor of 2, you actually added 18, so subtract 18 outside: 2 times the quantity x plus 3 squared minus 17.
For solving rather than rewriting, completing the square ends with a square root of both sides, which is where the plus or minus enters. From the quantity x plus 4 squared equals 25 you get x plus 4 equals 5 or negative 5, so x equals 1 or x equals negative 9. Forgetting the negative branch loses one of the two solutions every time.