We are tasked with finding the value of the expression:

$\frac{a^2}{bc} + \frac{b^2}{ac} + \frac{c^2}{ab}$

given that $a + b + c = 0$.

Step 1: Simplify the given expression

We can rewrite the expression as:

$\frac{a^2}{bc} + \frac{b^2}{ac} + \frac{c^2}{ab} = \frac{a^3 + b^3 + c^3}{abc}$

Step 2: Use the identity for $a^3 + b^3 + c^3$

We know that for $a + b + c = 0$, the identity for $a^3 + b^3 + c^3$ simplifies as follows:

$a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$

Since $a + b + c = 0$, this identity simplifies to:

$a^3 + b^3 + c^3 = 3abc$

Step 3: Substitute this into the expression

Now substitute $a^3 + b^3 + c^3 = 3abc$ into the original expression:

$\frac{a^3 + b^3 + c^3}{abc} = \frac{3abc}{abc} = 3$

Thus, the value of the expression is:

$\boxed{3}$

Would you like more details or further clarification?

Here are some related questions:

1. How does the identity for $a^3 + b^3 + c^3$ change if $a + b + c \neq 0$?
2. What is the geometric interpretation of the equation $a + b + c = 0$?
3. How do similar identities simplify expressions involving cubes of sums?
4. How can we extend this concept to higher powers, such as $a^4 + b^4 + c^4$?
5. Can you derive the same result using a different method, such as factorization?

Tip: Always check for known algebraic identities when simplifying complex expressions—this can often save a lot of time and effort!