If a+b+c=0 then prove that a^4+b^4+c^4=2(a^2b^2+b^2c^2+c^2a^2)

To prove the given statement, we can use the identity that relates the sum of fourth powers to the product of squared terms. Here's how to do it step by step:

1. Start with the given equation: a + b + c = 0.

2. Square the given equation to eliminate the sum: (a + b + c)^2 = 0^2.

Simplifying the left-hand side: a^2 + 2ab + 2ac + b^2 + 2bc + c^2 = 0.

3. Expand the equation: a^2 + b^2 + c^2 + 2ab + 2ac + 2bc = 0.

4. Rearrange the terms: a^2 + b^2 + c^2 = -2ab - 2ac - 2bc.

5. Square the equation again to eliminate the sums: (a^2 + b^2 + c^2)^2 = (-2ab - 2ac - 2bc)^2.

Expanding both sides: a^4 + b^4 + c^4 + 2a^2b^2 + 2a^2c^2 + 2b^2c^2 = 4a^2b^2 + 4a^2c^2 + 4b^2c^2 + 4ab^3 + 4ac^3 + 4bc^3.

6. Simplify the equation: a^4 + b^4 + c^4 + 2a^2b^2 + 2a^2c^2 + 2b^2c^2 = 4(a^2b^2 + a^2c^2 + b^2c^2) + 4(ab^3 + ac^3 + bc^3).

7. Notice that because a + b + c = 0, we can rewrite the equation as: a^4 + b^4 + c^4 + 2a^2b^2 + 2a^2c^2 + 2b^2c^2 = 4(a^2b^2 + a^2c^2 + b^2c^2).

Since a + b + c = 0, we can substitute -c for a and b in the equation to get: a^4 + b^4 + c^4 + 2a^2b^2 + 2a^2c^2 + 2b^2c^2 = 4(a^2b^2 + a^2c^2 + b^2c^2).

8. Finally, we can simplify further: a^4 + b^4 + c^4 = 2(a^2b^2 + b^2c^2 + c^2a^2).

Hence, we have proved that if a + b + c = 0, then a^4 + b^4 + c^4 = 2(a^2b^2 + b^2c^2 + c^2a^2).