Let's solve the expression step-by-step.

We are given:

$\left( x^2 - yz \right)^3 + \left( y^2 - zx \right)^3 + \left( z^2 - xy \right)^3 - 3 \left( x^2 - yz \right) \left( y^2 - zx \right) \left( z^2 - xy \right)$

Step 1: Use an identity for cubic expressions

We notice that the given expression resembles a well-known identity for the sum of cubes:

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

Let's define:

$a = x^2 - yz, \quad b = y^2 - zx, \quad c = z^2 - xy$

Thus, the expression becomes:

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

Step 2: Calculate $a + b + c$

Now, compute $a + b + c$:

$a + b + c = (x^2 - yz) + (y^2 - zx) + (z^2 - xy)$

Simplifying:

$a + b + c = x^2 + y^2 + z^2 - (yz + zx + xy)$

Thus, we have:

$a + b + c = x^2 + y^2 + z^2 - (xy + yz + zx)$

Step 3: Check if the expression can be a perfect square

The expression involves a factor $(a + b + c)$, and if $a + b + c = 0$, the equation simplifies to:

$a^3 + b^3 + c^3 - 3abc = 0$

In this case, we would be left with a factorization:

$(x^2 - yz) + (y^2 - zx) + (z^2 - xy) = 0$

Step 4: Solve the equation for roots

Now, let's solve the equation $x^2 + y^2 + z^2 - (xy + yz + zx) = 0$. This is a symmetric equation in $x$, $y$, and $z$. A well-known solution to this symmetric equation is $x = y = z$.

Step 5: Find the roots

Let $x = y = z$. Substituting this into the expression:

$a = b = c = x^2 - x^2 = 0$

Thus, the entire expression becomes $0$, which is trivially a perfect square.

Conclusion

The expression is indeed a perfect square when $x = y = z$.

Further questions for exploration:

1. How would you prove that $x = y = z$ is the only solution to the equation?
2. What happens if we assume $x \neq y \neq z$? How can we analyze the equation?
3. How does the symmetry of the equation help simplify the analysis?
4. Can this result be generalized to higher-degree polynomials with symmetric terms?
5. How can we apply this identity in more complex cubic equations?

Tip:

When dealing with symmetric equations, consider simplifying the problem by assuming equal values for the variables (e.g., $x = y = z$) and check if this leads to a solution.