factor by grouping number method using FOIL

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We want to construct an expression that when simplified gives 2x^2 + 5x - 3, that look like (ax+b)(cx+d), and we need to find what a, b, c and d are.

By FOIL, the first terms product F: is a*c*x^2

the second term product O: is a*d*x

the third term product I: is b*c*x

and the last product is L: b*d.

We need to match that with 2*x^2 + 5x - 3, which means that:

a*c = 2 [x^2 terms]

a*d + b*c = 5 [x terms]

b*d = -3 [constant terms]

We get that b=-1, d = 3, a = 2, c =1 or

(2x-1)(x+3).

Just to confirm: 2x*x +6*x -1*x -3 = 2x^2 + 5x - 3, as desired.

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