The first two terms don't have anything in common. Let's switch the middle two terms. If you can, see if you can do it without waking them. Those guys have had a long day and are all tuckered out.
y2 + 7y + 35x + 5xy
and now we can do our thing. The first two terms have a y in common, and the second two terms have 5x in common. We factor for
y(y + 7) + 5x(7 + y).
Since addition is commutative y + 7 = 7 + y, the previous line can be written as
y(y + 7) + 5x(y + 7).
Then we pull out the common factor of (y + 7) and get
(y + 5x)(y + 7).
Sometimes we have a choice of factors, positive or negative, and need to pick the negative one. We hate to be a negative Nancy, but there are instances in which it makes more sense than to be a positive Peter.
Factor 2x2 + 14x - 3x - 21 by grouping.
The first two terms have 2x in common, which we can factor out to find
2x(x + 7) - 3x - 21.
For the second two terms, we could factor out 3 or we could factor out - 3. Whoa! Rein in your pony there, cowboy. If we factor out 3, we'll find
2x(x + 7) + 3(-x - 7).
Those two terms don't have a common factor. Instead, if we had factored out - 3 we would get
2x(x + 7) - 3(x + 7)
and we could pull out (x + 7) to see that
(2x - 3)(x + 7).
Factor by grouping: 2x2 - 9x - 35.
Here we need two numbers whose product is -70 and whose sum is -9 (don't forget those negative signs). To have a product of -70, exactly one of the numbers must be negative. We don't care which one, as long as somebody volunteers.
Numbers that have a product of 70:
Since -14 + 5 = -9, we can stop looking at factors of 70 now. Good, because our eyes were starting to zone out a bit.