Remember that 5x^2 is the result of "first times first" multiplication, 18x is the sum of "outside times outside" and "inside times inside" multiplications, and -35 is the result of "last times last" multiplication.
To begin, think about the factor pairs whose product is 35.
1*35 and 5*7 are the only options.
Now consider the factor pair whose product is 35x^2.
1x*5x is the only one.
We know that the two "first" terms in the factorization will be 1x and 5x. We can easily discard 1*35 as the factor pair to obtain the product of 35, because of the large sums that would result from adding the outside & inside products.
Therefore we must decide on the proper placement of the 5 and 7, along with one + and one - & the effect of the 5x, to get the outside & inside sum of 18x.
(x + 7)(5x - 5) does not work, because the "outside times outside" multiplication results in -5x & the "inside times inside" multiplication results in 35x. The sum of these two products is 30x. Try again.
(x + 5)(5x - 7) works. Do FOIL to verify:
(x + 5)(5x - 7)
5x^2 - 7x + 25x - 35
5x^2 + 18x - 35
This verifies that the correct factorization is
(x + 5)(5x - 7).
