Calculate the following expressions: a) (x+3)^2-(x-3)^2+(x+3)(x-3) b) (x^2+x+1)-(x+1)^2 c) (x+2)(x^2-2x+4)-(x-3)(x^2+3x+9)

a) E(x)= (x+3)^2-(x-3)^2+(x+3)(x-3)

(x+3)^2=x^2+2*3*x+9 

(x-3)^2=x^2-2*3*x+9

(x+3)(x-3) = x^2-9

E(x)= x^2+2*3*x+9 – (x^2-2*3*x+9) + x^2-9

E(x)= x^2+2*3*x+9 – x^2+2*3*x-9 + x^2-9

After the process of reducing similar terms, the expression will become:

 E(x)= x^2 + 12x -9

b) E(x)= (x^2+x+1)-(x+1)^2

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

E(x)= (x^2+x+1)-( x^2 +2x + 1)

If we are substituting x^2 +2x + 1 with a letter, t, the expression will become

E(t)=t-t

E(t)=0

So, the expression is not depending on the variable. 

c) E(x)= (x+2)(x^2-2x+4)-(x-3)(x^2+3x+9)

E(x)=x*(x^2-2x+4)+2*(x^2-2x+4)-x*(x^2+3x+9)+3*(x^2+3x+9)

E(x)= x^3-2x^2+4x+2x^2-4x+8- x^3-3x^2-9x+3x^2+9x+27

After the process of reducing similar terms, the expression will become:

E(x)=8+27

E(x)=35

This expression does not depend on any variable, also.