$3{x}^{2}+5x+2$

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Split the second term in

3{x}^{2}+5x+2

into two terms.

Multiply the coefficient of the first term by the constant term.

3\times 2=6

Ask: Which two numbers add up to

5

and multiply to

6

3

and

2

Split

5x

as the sum of

3x

and

2x

3{x}^{2}+3x+2x+2

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3{x}^{2}+3x+2x+2

Factor out common terms in the first two terms, then in the last two terms.

3x(x+1)+2(x+1)

Factor out the common term

x+1

(x+1)(3x+2)

Done

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3x2+5x+23{x}^{2}+5x+23x2+5x+2