Step by step
The question
Question: factor x² − 2x − 15.
Factor means: write it as a times, with brackets.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Find two numbers
First, see how two brackets open.
(x + 2)(x + 3) gives x² + 3x + 2x + 6.
That is x² + 5x + 6.
2 and 3 multiply to make the last number, 6.
And they add to make the middle number, 5.
Factoring goes backwards.
We need two numbers.
They multiply to −15, and they add to −2.
They multiply to a minus number, so one number is plus and one is minus.
Only plus times minus gives a minus.
They add to a minus number, so the bigger one is minus.
When we add a plus and a minus, the bigger one wins.
We list the pairs that make 15, smallest first.
One is minus, so adding them is a take away.
We take the smaller from the bigger.
Without the signs, we need 2.
3 and 5 give 2.
Yes!
With the signs: 3 and −5.
The two numbers go in the brackets.
So we get (x + 3)(x − 5).
2 Check
Let's check: we multiply it back out.
Each part of the first bracket times each part of the second.
We get x² − 2x − 15 again.
So (x + 3)(x − 5) is right!
3 Pick the choice
Our answer is (x + 3)(x − 5).
That is choice C.
4 Why not the others
A is (x + 5)(x − 3).
Wrong: 5 + (−3) = +2, not −2.
B is (x − 15)(x + 1).
Wrong: −15 + 1 = −14, not −2.
D is (x − 5)(x − 3).
Wrong: (−5) × (−3) = +15, not −15.
Let's look back.
First: find two numbers.
Then: check.
Then: pick the choice.
Last: why not the others.