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To prove the given expressions involving the variables \( a \), \( b \), \( c \), and \( d \), we need to analyze each part step-by-step and apply algebraic identities and manipulations.

### Part (a)
We start with:
\[
a^2b^2 - b^2c^2 - (ca + bd)^2 - (ae + bc)^2
\]
This looks like it can be simplified by recognizing factors or applying the difference of squares. Without specific numerical values or further context, we can hypothesize that it reduces to a recognizable pattern in symmetric sums.

### Part (b)
Next, we address:
\[
h(a^3 + b^3 + c^3 - abc) = (a + b + c)^2 - [(a - b)^2 + (b - c)^2 + (c - a)^2]
\]
We recognize that \( h \) often represents a symmetric polynomial or a determinant-related condition. The identity relates sums of cubes to squared differences, illustrating symmetrical properties in algebra.

### Part (c)
For:
\[
(ca + b)(b + c) - (cab + h)^2 = (ca + b)^2 + a(b + c)^2 + b(ca + a)^2
\]
This involves distributive property and may require expansion to demonstrate equality or a rearrangement of terms. Again, recognizing symmetric behavior in \( a, b, c \) will help condense the expression into something manageable.

### Part (d)
The identity provided:
\[
(c + d)^3 = a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a)
\]
Can be approached by applying the binomial theorem and the relation between cubic polynomials and symmetric sums.

### Additional Problem
Finally, the expression:
\[
x^2 + y^2 + z^2 - nxy + yz + zx
\]
Requires substitution or quadratic completion to resolve for specific roots or the minimum value according to the quadratic forms.

### General Approach
1. Use polynomial identities (e.g., sum of cubes, difference of squares).
2. Expand and combine like terms systematically.
3. Apply symmetry when appropriate to conclude each step.

Each part may require deeper algebraic inspection to arrive at a final proof or conclusion, ideally grounded in known mathematical identities. If specific results are desired, numerical examples or structured proofs could be drafted for clarity. Would you like to pursue a specific part in more detail?