$ ext{If } a + b + c = 0 ext{, then } \frac{a^4 + b^4 + c^4}{a^2b^2 + b^2c^2 + c^2a^2} ext{ is equal to :}$

Q: $\text{If } a + b + c = 0 \text{, then } \frac{a^4 + b^4 + c^4}{a^2b^2 + b^2c^2 + c^2a^2} \text{ is equal to :}$

$\text{If } a + b + c = 0 \text{, then } \frac{a^4 + b^4 + c^4}{a^2b^2 + b^2c^2 + c^2a^2} \text{ is equal to :}$

Question

$	ext{If } a + b + c = 0 	ext{, then } \frac{a^4 + b^4 + c^4}{a^2b^2 + b^2c^2 + c^2a^2} 	ext{ is equal to :}$

Accepted Answer

To solve the problem, we need to evaluate the expression $\frac{a^4 + b^4 + c^4}{a^2b^2 + b^2c^2 + c^2a^2}$ given that $a + b + c = 0$.

### Step-by-step Explanation:

1. **Substitute $c = -(a + b)$:**
   Since $a + b + c = 0$, we can express $c$ as $c = -(a + b)$.

2. **Expand $a^4 + b^4 + c^4$:**
   Substitute $c = -(a + b)$ into the expression:
   $$
   c^4 = (-(a + b))^4 = (a + b)^4
   $$
   Expanding $(a + b)^4$ using the binomial theorem:
   $$
   (a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4
   $$
   Therefore:
   $$
   a^4 + b^4 + c^4 = a^4 + b^4 + (a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4)
   $$
   Simplifying:
   $$
   = 2a^4 + 2b^4 + 4a^3b + 6a^2b^2 + 4ab^3
   $$

3. **Expand $a^2b^2 + b^2c^2 + c^2a^2$:**
   Substitute $c = -(a + b)$:
   $$
   c^2 = (-(a + b))^2 = (a + b)^2 = a^2 + 2ab + b^2
   $$
   Therefore:
   $$
   b^2c^2 = b^2(a^2 + 2ab + b^2) = b^2a^2 + 2a^2b^2 + b^4
   $$
   $$
   c^2a^2 = (a^2 + 2ab + b^2)a^2 = a^4 + 2a^3b + a^2b^2
   $$
   Summing these:
   $$
   a^2b^2 + b^2c^2 + c^2a^2 = a^2b^2 + (b^2a^2 + 2a^2b^2 + b^4) + (a^4 + 2a^3b + a^2b^2)
   $$
   Simplifying:
   $$
   = a^4 + b^4 + 3a^2b^2 + 2a^3b + 2ab^3
   $$

4. **Simplify the Expression:**
   Now, substitute the expanded forms into the original expression:
   $$
   \frac{2a^4 + 2b^4 + 4a^3b + 6a^2b^2 + 4ab^3}{a^4 + b^4 + 3a^2b^2 + 2a^3b + 2ab^3}
   $$
   Factor out common terms:
   $$
   = \frac{2(a^4 + b^4 + 2a^3b + 3a^2b^2 + 2ab^3)}{a^4 + b^4 + 2a^3b + 3a^2b^2 + 2ab^3}
   $$
   Simplifying gives:
   $$
   = 2
   $$

### Conclusion:

The expression simplifies to 2, which confirms that Option 1 is correct.

### Explanation for Incorrect Options:

- **Option 2 (1):** The simplification clearly shows the result is 2, not 1.
- **Option 3 (0):** The expression does not simplify to zero; the terms do not cancel out completely.
- **Option 4 (-2):** The expression does not yield a negative result; all terms are positive or zero.

Thus, the correct answer is indeed Option 1, which is 2.