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  3 x 4 = 12;  3 and 4 are factors of 12.  6 and 2 are factors of 12 as are 1 and 12. If you are asked to factor 12 you write 12 = 3 x 4 or 12 = 2 x 6 or 12 = 1 x 12 

In other words you are being asked: What did you multiply together to obtain 12?

In algebra:

3(r + 2) = 3r + 6    3 and (r + 2) are factors of 3r + 6

When you are asked to factor 3r + 6 you are being asked to convert 3r + 6 to 3(r + 2).

In other words you are being asked: What did you multiply together to obtain 3r + 6?

Factor this problem;   4p + 8 =

4p + 8 = 4p + 4(2)  4 and p are factors of each other and 4 and 2 are factors of each other. The largest common factor of 4p and 4(2) is 4. Therefore 4p + 8 = 4(p + 2)   Are there other common factors of 4p + 8?  Yes. 4p + 8 = 2(2p + 4)

However; When asked to factor we want the largest common factor which in this case is 4.

Factor 4t + 7t2   =  2(2)t + 7t(t) = t(4 + 7t)  t is the only common factor of 4t and 7t2.

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