(a)
(x + x2 + x3 + x4)2 = x2 +2x3+3x4+4x5+3x6+2x7+x8
(b)
Scores
2
3
4
5
6
7
8
Frequencies
1
2
3
4
3
2
1
Relative frequencies
0.0625
0.125
0.1875
0.25
0.1875
0.125
0.0625

(c) The coefficients and the frequencies are the same.

(d) Here is where users can experiment with Mike's Probability Environment. (e) Possible answers 1, 3, 3, 5 and 1, 2, 2, 3 or 0, 2, 2, 4 and 2, 3, 3, 4 or 2, 4, 4, 6 and 0, 1, 1, 2

(f)The spinner-pairs in (e) correspond to the three factorisations below:
(x + x2 + x3 + x4)2 = x2 (1 + x)2 (1+x2)2 .

[x(1+x2)2][x(1+x)2]
= [x + 2x3 + x5][x + 2x2 + x3]
(1)
[(1+x2)2][x2(1+x)2]
= [1 + 2x2 + x4][x2 + 2x3 + x4]
(2)
[x2(1+x2)2][(1+x)2]
= [x2 + 2x4 + x6][1 + 2x + x2] .
(3)
(g) Also
(x + x2 + x3 + x4)2
= x2 (1 + x)2 (1+x2)2
= [x2(1+x)][(1+x)(1+x2)2]
=[x2 +x3][1+x+2x2 +2x3+x4+x5]
suggests a 2-spinner labelled 2, 3 and an 8-spinner labelled 0, 1, 2, 2, 3, 3, 4, 5.