There are two possibilities. One is that this factors nicely (we know it does in this case). The other is that it doesn't factor nicely (givings us irrational answers like x = 3.584538257389025 etc.). The steps below should work if the equation factors nicely, but won't work if the answers are irrational / not friendly.
Note: Once you get used to it, you won't have to follow all of these steps and do all of this stuff every time. You'll get used to it, like how you don't have to do a multiplication problem on paper to know that 10 * 10 = 100. Here's the step by step procedure though.
6x^2 + 7x - 3 = 0
We're looking for something like (?x + ??)(???x + ????)
The number on the x^2 is 6, so the numbers in front the x's (? and ???) have to multiply together and make 6. That means ? and ??? should be 1 and 6, 6 and 1, 2 and 3, or 3 and 2. That means we should have something like:
(x + ??)(6x + ????) or (2x + ??)(3x + ????)
Now look at the number without an x in the equation: -3. The only way to factor 3 is 1 and 3. But, since we're starting with -3, the factors can be 1 and -3 or -1 and 3. That means we now have something like this:
(x + 1)(6x - 3) or (2x + 1)(3x - 3)
(x - 1)(6x + 3) or (2x - 1)(3x + 3)
(x + 3)(6x - 1) or (2x + 3)(3x - 1)
(x - 3)(6x + 1) or (2x - 3)(3x + 1)
So that's all eight possible combinations. Which one is right?
In the equation we have 7x in the middle. We have to get a 7 to sit on the x.
Look at (x + 1)(6x - 3). The number sitting on the x in the equation is made by x * -3 + 1 * 6x = -3x + 6x = 3x. That didn't make 7 to sit on the x, so we know (x + 1)(6x - 3) is wrong.
(x + 1)(6x - 3) or (2x + 1)(3x - 3)
(x - 1)(6x + 3) or (2x - 1)(3x + 3)
(x + 3)(6x - 1) or (2x + 3)(3x - 1)
(x - 3)(6x + 1) or (2x - 3)(3x + 1)
Let's go through the other possibilities and see what we get
(x + 1)(6x - 3): 3x or (2x + 1)(3x - 3): -3x
(x - 1)(6x + 3): -3x or (2x - 1)(3x + 3): 3x
(x + 3)(6x - 1): 17x or (2x + 3)(3x - 1): 7x
(x - 3)(6x + 1): -17x or (2x - 3)(3x + 1): 3x
The only possibility that makes a 7 to sit on the x is (2x + 3)(3x - 1), so that's our answer.
6x^2 + 7x - 3 = (2x + 3)(3x - 1)