When factoring, we look for something called a common factor; in other words, something that appears in all of the terms. Right off the bat, we can see that all of our coefficients (the numbers next to the xs) are even - so 2 is a common factor. In addition, all of our terms contain an x. Putting it all together, our first common factor is 2x:
2x(x^2+5x+6)
If we were to distribute the 2x, we would get 2x^3+10x^2+12x, which is our original problem.
Now, we proceed to factoring the quadratic equation x^2+5x+6. We do this by looking for two numbers that add to 5 and multiply to 6; these numbers are 3 and 2. If you're wondering why: when we factor a quadratic like this one, we want it in the form (x+a)(x+b), where a and b multiply to the constant term and add to the first-degree term (in this case, our constant term is 6 and our first degree term is 5x). Here, we have a = 3 and b = 2 (a = 2 and b = 3 works too). So, the factored form of the quadratic equation x^2+5x+6 = (x+3)(x+2). Back to the main problem.
Now that we have our quadratic completely factored, we can finish up. We simply replace our quadratic with the factored version, like this:
2x(x+3)(x+2)
Since we can't simplify this any further, we can say that y = 2x^3+10x^2+12x in factored form is y = 2x(x+3)(x+2).
