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Well since we're adding two numbers that result in a negative, and a product that is negative, we're looking for a positive and a negative number.
-24
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1 is prime
-12 x 2 = -24 = -10
-8 x 3 = -24 = -5
-6 x 4 = -24 = -2
5 doesn't make 24
4 x -6 = -24 = -2 (Uh oh, repeating multiplication)
Once you begin repeating multiplication you've found all possibilities that could result in the sum of -6 and the product of -24. However, since you've found no numbers to result in your desired sum/product, the trinomial in which you are factoring is prime.
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