Example 5:  Factor  6x² – 23x + 20

Solution: We concentrate on 6x² and 20 first.

6x² could result from 6x times x or from 3x times 2x.

Let's try 3x times 2x and worry about the 20:

6x² – 23x + 20 = (3x         )(2x         )

20 could be 2 times 10, 10 times 2, 4 times 5, 5 times 4, 1 times 20, or 20 times 1.  Since the middle term is negative, both factors of 20 should be negative.

Let's try –2 times –10:

(3x – 2)(2x – 10) = 6x² – 34x + 20

The middles add to –34x which is wrong.

So we try –5 times –4:

(3x – 5)(2x – 4) = 6x² – 24x + 20

Close, but not right.  The middles should add to –23x, not –24x.

Try –4 times –5:

(3x – 4)(2x – 5) = 6x² – 23x + 20

This is right, so we know 6x² – 23x + 20 factors as (3x – 4)(2x – 5)

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