Let the numbers be a-2d, a-d, a and a+d.

Their sum is

a-2d+ a-d+ a + a+d = 20

4a-2d = 20

2a - d = 10 …(1)

Now, Sum of square is

(a - 2d)² + (a - d)² + a² + (a + d)² = 180

a² – 4ad + 4d^2 + a^2 – 2ad + d^2 + a^2 + a^2 + 2ad + d^2 = 180

a^2 – 4ad + 4d^2 + a^2 + d^2 + a^2 + a^2 + d^2 = 180

4a2 – 4ad + 6d2 = 180

2a^2 – 2ad + 3d^2 = 90 …(2)

From (1)

d = 2a -10.

Put that in (2) to get

2a^2 -2a(2a–10) + 3(2a–10)^2 = 90, or

2a^2 + 20a - 4a^2 + 3(4a^2 - 40a + 100) = 90

2a^2+20a-4a^2+300–120a+12a^2 = 90

10a^2–100a+210 = 0

a^2–10a+21=0

(a-3)(a-7) = 0

Hence a = 3 or 7 and the corresponding value of d from (1)

d=2a - 10 is -4 or 4.

So the 4 terms of the AP are 3-[2*(-4)] = 11, 7, 3 and -1, or

-1, 3, 7 and 11.

Check:

11 + 7 + 3 – 1 = 20. Correct.

1^2+3^2+7^2+11^2 = 1+9+49+121 = 180. Correct.