Actually I figured out part c. This is what I have:

x^3 = x + 1

x^4 = x*x^3 = x*(x + 1) = x^2 + x

x^5 = x*x^4 = x*(x^2 + x) = x^3 + x^2 = x^2 + x + 1

x^6 = x*x^5 = x*(x^2 + x + 1) = x^3 + x^2 + x = (x + 1) + x^2 + x = x^2 + 2x + 1 = x^2 + 1

x^7 = x*x^6 = x*(x^2 + 1) = x^3 + x = (x + 1) + x = 2x + 1 = 1

So,

<x> = { x + 1, x^2 + x, x^2 + x + 1, x^2 + 1, 1 }