Multiplication, backwards
You already know expanding: using the distributive property, 3(x + 2) becomes 3x + 6. [[factoring|Factoring]] is that same step run in reverse — you start with 3x + 6 and write it back as 3(x + 2), a product of factors. Nothing about the expression changes value; you are just dressing it in a more useful form. A factored expression is handy because a product equal to zero tells you something a sum never could, which is the punchline of this whole track.
To factor is the verb; the factoring result should always be checked by multiplying back out. If re-expanding does not return your original polynomial, you made an error somewhere. This single habit — multiply your answer back — will catch most mistakes before they cost you anything.
Finding the greatest common factor
The first thing to try on any polynomial is the [[greatest-common-factor|greatest common factor]] of its terms, often written GCF. For the numbers, take the largest integer that divides every coefficient. For each variable, take the lowest power that appears in every term. Multiply those together to get the common monomial factor, then divide it out of each term.
Factor 12x^3 + 18x^2 - 30x Coefficients 12, 18, 30 -> GCF = 6 Variable part: lowest power of x is x^1 So the GCF of all terms is 6x 12x^3 / 6x = 2x^2 18x^2 / 6x = 3x -30x / 6x = -5 12x^3 + 18x^2 - 30x = 6x(2x^2 + 3x - 5) Check: 6x * 2x^2 = 12x^3, 6x * 3x = 18x^2, 6x * (-5) = -30x ✓