Calculate (x+y)^2 + (x+y+1)^2 x^2003-1 - (x -1)(x^2002+ ...+ x+1) =...

1) To calculate the first expression, we'll expand the
squares, using the formulas:

(a+b)^2 = a^2 + 2ab +
b^2

(a+b+c)^2 = a^2 + b^2 + c^2 +
2(ab+ac+bc)

(x+y)^2 = x^2 + 2xy +
y^2

(x+y+1)^2 = x^2 + y^2 + 1 + 2(xy + x +
y)

We'll re-write the expressions substituting the
squares:

E (x,y) = x^2 + 2xy + y^2 + x^2 + y^2 + 1 + 2(xy +
x + y) - (3x^2 + 4xy) + (y-1)

Now, we'll remove the
brackets and combine like terms:

E(x,y) = 2x^2 + 2y^2 + 2xy
+ 1 + 2xy + 2x + 2y - 3x^2 - 4xy + y - 1

We'll combine and
eliminate like terms:

E(x,y) = 2x^2 + 2y^2+ 2x + 2y - 3x^2
+ y

E(x,y) = -x^2 + 2y^2 + 2x +
3y

2) Now, we'll calculate the
expression:

x^2003 - 1 - (x -1)(x^2002+ ...+ x+1)
=...

We'll use the
formula:

x^n - 1 = (x-1)(x^(n-1) + x^(n-2) + ......... + x
+ 1)

We'll substitute n by
2003

x^2003 - 1 = (x-1)(x^2002 + x^2001 + ........ + x +
1)

We'll re-write the
expression:

E (x) = (x-1)(x^2002 + x^2001 + ........ + x +
1) - (x -1)(x^2002+ ...+ x+1) 

We'll eliminate like
terms:

 E (x) =
0