factor the polynomials using whatever strategy seems appropriate.
12m^2n^2-8mn+1
Just by inspection,
(2mn-1)(6mn-1)
How did you get that?
Think of it like factoring
12x^2 - 8x + 1
or
x^2 - 8x + 12
You want factors of 12 that add to 8
That'd be 6 and 2.
It takes some experience to get quick at factoring by inspection. Do lots of these till you don't have to spend a lot of time guessing.
If the middle term is small, and the last term is negative, you want factors that are close to each other:
(x^2-4x+96) = (x-12)(x+8)
If the middle term is positive, it's easier to imagine factors that add up.
Of course, if you are faced with something like
(12x^2 - 29x + 60)
you will have lots of possible combinations.
To factor the polynomial 12m^2n^2 - 8mn + 1, we can use the strategy of factoring by grouping.
Step 1: Look for a common factor among all terms.
In this case, there is no common factor among all the terms.
Step 2: Look for a common factor among pairs of terms.
In this case, we can factor 4mn from the first two terms:
12m^2n^2 - 8mn = 4mn(3m^2n - 2)
Step 3: Factor out the remaining terms.
The expression becomes:
4mn(3m^2n - 2) + 1
Step 4: The polynomial is now fully factored.
The factored form of the polynomial 12m^2n^2 - 8mn + 1 is 4mn(3m^2n - 2) + 1.