How do you FACTOR this expression

x^2 + 4x + 4

1. I'm NOT trying to solve for x.
2. I KNOW the answer is (x + 2)^2, so I don't want the answer. What I DO want is a step by step process of how to get to that answer.

With the same rules, please also tell me HOW (with a step by step process) to factor this: x^2 -2x +1
Again, I do NOT just want the answer!

You have to use the rule of FOIL. Multiply the First two, the Outer two, the Inner two, and the Last two working backwards.

x^2 -2x + 1

You'll always have (x plus/minus a number)(x plus/minus a number). Since you know that it is just x^2, that is simply X*X which will give you the below info:
(x +/- ?)(x +/- ?)
Since you know the last part is "+1", you can get there either by multiplying 1*1 or -1*-1. In order to determine which one of those two it is, you have to look at the middle data which is -2x. If you used 1*1, that wouldn't give you -2x so yo have to use the -1*-1

(x-1)(x-1) is your answer or perhaps (x-1)^2, to check that you'd have:
F: x*x = x^2
O: x * -1 = -1x
I: -1 * x = -1x
L: -1 * -1 = 1

X^2 -1x -1x +1 x^2 -2x +1



Same logic applies to y^2 + 4y + 4
You know the +4 could either be (1&4),(2&2),(-1&-4), or (-2&-2). Looking at the middle component, you know it's not either of the negatives, and you also can determine that 1y+4y=5y; so it is the (2&2) option for the Last two.
(y+2)(y+2) is your answer or perhaps (y+2)^2

This method works only when leading coefficient (coefficient of x²) = 1 :

x² + 4x + 4

To factor in form (x + a) (x + b) you need to find two numbers (a and b)
whose product = constant factor = 4 and
whose sum = coefficient of x = 4

2 * 2 = 4
2 + 2 = 4

So a = 2, b = 2

x² + 4x + 4 = (x + 2) (x + 2) = (x + 2)²

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Alternatively, you could factor in pairs: