Factoring x² + bx − c

The question

x² + 8x − 345

multiply to−345· add to8

−345 < 0→one +, one −

1·3453·1155·6915·23

23and−15

x² + 8x − 345=(x + 23)(x − 15)

Factoring this means finding two numbers to put in (x + ▢)(x + ▢). They multiply to −345, the last term, and add to 8, the number in front of x.
−345 is negative, and a product is only negative when the two numbers have opposite signs. So one is positive and one is negative, and the bigger of the two takes the sign of 8.
Now list the pairs that multiply to 345. There are only 4: 1 and 345, 3 and 115, 5 and 69, 15 and 23.
Which of them adds to 8? 23 × −15 = −345 and 23 + −15 = 8. So the two numbers are 23 and −15.
Put them in the brackets: x² + 8x − 345 = (x + 23)(x − 15). Multiply back out to check.

How it works.

01
Factoring this means finding two numbers to put in (x + ▢)(x + ▢). They multiply to −345, the last term, and add to 8, the number in front of x.
02
−345 is negative, and a product is only negative when the two numbers have opposite signs. So one is positive and one is negative, and the bigger of the two takes the sign of 8.
03
Now list the pairs that multiply to 345. There are only 4: 1 and 345, 3 and 115, 5 and 69, 15 and 23.
04
Which of them adds to 8? 23 × −15 = −345 and 23 + −15 = 8. So the two numbers are 23 and −15.
05
Put them in the brackets: x² + 8x − 345 = (x + 23)(x − 15). Multiply back out to check.