The last one first because I think it is the easiest. any time we have adifference of squaresex.(x^2 - y^2) it will factor as (x+y)* (x-y)

ex. x^2 - 4 is also thedifferance of squares. x^2 = x*x and 4 = 2*2

So it will factor (x+2)*(x-2)

In your example 4x^2-81y^2 notice that 4=2*2 and x^2=x*x and 81=9*9 and y^2=y*y.

Do you see that every number is a square?

So we will take 2x from the first term and 9y from the last term and put it into the form (x+y)*(x-y) and we get (2x+9y)*(2x-9y) So this is called aDifference Of Squares

The first problem is called aperfect square. Notice that the first and last term are squares. The first term 100b^2 = 10*10*b*b and the last term is 81=9*9.

But notice that the middle term is equal to 10*b*9*2 or 180b

Let me show you

(10b+9)^2= (10b +9)*(10b+9)

UsingFOILwe get

First 10b*10b =100b^2

Outside 10b*9 = 90b (please notice that we will do this one more time)

Inside 9*10b= 90b

Last 9*9=81

The inside and outside terms are added together and in a Perfect Square both the inside and outside terms are the same

So when we factor a Perfect Square (sometimes refered to as a Trinomial Square) the general form is

A^2 +2AB +B^2 = (A+B)^2 and

A^2 -2AB +B^2 = (A-B)^2

So finally back to your problem 100b^2 + 180b +81 where 100=10*10 and b^2=b*b and 180b= 2*90b and 81=9*9 we get (10b+9)^2

Hope this helps

One more thingFOILmeans First Outside Inside Last, it describes the order in which we multiply these factors ex (a+b)*(c+d)

1. miltiply theFirst terms a and c

2. multiply theOutside terms a and d

3. multiply theInside terms b and c

4. miltiply theLast terms b and d