Q:  Expand:  ((x+4y)+3)²

A:  There are many ways to do this problem, but I am going to pick the one I like best:

((x+4y)+3)² = (x+4y+3)²

[that extra set of parentheses was not needed!]

So,

(x+4y+3)² = (x+4y+3)(x+4y+3)

Now, we just distribute out the right side.  Remember, everyone touches everyone (but not the people you came with)…. If what I said makes no sense, read some of the older examples of distributing polynomials 🙂

However, follow along (the colors are just meant to help you figure out where my numbers came from):

(x+4y+3)(x+4y+3) = x*x + x*4y + x*3 + 4y*x + 4y*4y + 4y*3 + 3*x + 3*4y + 3*3

= x² + 4xy + 3x + 4xy + 16y² + 12y + 3x + 12y + 9

Now, combine like terms:

= x² + 8xy + 6x + 24y + 16y² + 9