(X*3) (x*1) = x^2 + 4x +3. How is the answer found? What are the steps to algebraic factoring?

Not sure if that's really what you mean, but taken literally as you have it, we have:

(x * 3)(x * 1) = x² + 4x + 3

Simplify the left side:

3x(x) = x² + 4x + 3
3x² = x² + 4x + 3

Now move everything to one side to put it into standard polynomial form:

2x² - 4x - 3 = 0

Now to solve this, I'll use complete the square. So I'll add 3 to both sides, then divide both sides by 2:

2x² - 4x = 3
x² - 2x = 3/2

Adding 1 to both sides to complete the square:

x² - 2x + 1 = 5/2

Factor, then square root:

(x - 1)² = 5/2
x - 1 = ± √(5/2)

Add 1 to both sides, then rationalize the denominator:

x = 1 ± √(10) / 2

You have two solutions.

I think it's (x+3)(x+1) = x² + 4x + 3

So I always think: ADD, MULTIPLY.
What two numbers add to 4?
What same two numbers as before also multiply to 3?

3 + 1 = 4.
3 * 1 = 3.

x² + 4x + 3 = (x + ?)(x + ?)
x² + 4x + 3 = (x + 3)(x + 1)

ADD to the x term. MULTIPLY to the constant.
Oh and importantly, this works only when the leading coefficient on x² is 1!

Assuming X = x
2x^2-4x-3 = 0
x = [4±√40]/4
x = 1±(1/2)√10

(x*3) (x*1) = x^2 + 4x +3
2x^2 - 4x - 3 = 0
2 (x - 1)^2 - 5 = 0
Solutions:
x = 1 - sqrt(5/2)
x = 1 + sqrt(5/2)