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

I'm not clear on what you are asking.

If your expression was:
(x + 3)(x + 1)

Then you can expand that using distributing or FOIL.

Distributing:
x(x + 1) + 3(x + 1)
= x² + x + 3x + 3
= x² + 4x + 3

FOIL:
First: x * x = x²
Outer: x * 1 = x
Inner: 3 * x = 3x
Last: 3 * 1 = 3

Add those up and combine the like terms in the middle:
x² + x + 3x + 3
= x² + 4x + 3

If you are doing the reverse, then you are looking for two numbers that multiply to be 3 but add to be 4. It should be pretty apparent that there is only one choice for two integers that multiply to be 3:
1 * 3 = 3

And from there we get a sum of 4.
1 + 3 = 4

So the answer is:
(x + 1)(x + 3)

You can write the binomials in either order, so (x + 3)(x + 1) is the same factoring.