2x² + 7x + 3
2x² + 7x + 3

1.  Multiply the blue 2 by the red 3, getting 6.
2.  Think of two positive integers which have product 6 and
    which have SUM (since the purple sign is +) equal to the
    green 7.  Such positive integers are 1 and 6 because 
    1×6 = 6 and 1+6 = 7
3.  Rewrite the green 7 using 1 and 6 

2x² + (1 + 6)x + 3

4. Remove the parentheses using the distributive principle.

2x² + 1x + 6x + 3

5. Factor x out of the first two terms and factor 3 out of
   the last two terms:

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

6. Factor the (2x + 1) out of both terms

(2x + 1)(x + 3)

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Another example:
6x² + x — 2
6x² + 1x — 2

1.  Multiply the blue 6 by the red 2, getting 12.
2.  Think of two positive integers which have product 12 and
    which have DIFFERENCE (since the purple sign is —) equal to the
    green 1.  Such positive integers are 4 and 3 because 
    4×3 = 12 and 4—3 = 1
3.  Rewrite the green 1 using 4 and 3 

6x² + (4 — 3)x — 2

4. Remove the parentheses using the distributive principle.

6x² + 4x — 3x — 2

5. Factor 2x out of the first two terms and factor -1 out of
   the last two terms:

2x(3x + 2) — 1(3x + 2)

6. Factor the (3x + 2) out of both terms

(3x + 2)(2x — 1)

Edwin