We are tasked with simplifying the expression:

\[
\frac{(6 + 6a^2 + 6h^2 - 12ah) - (6 + a^2)}{h}
\]

Let's break this down step by step.

### Step 1: Simplify the numerator
The numerator is:

\[
(6 + 6a^2 + 6h^2 - 12ah) - (6 + a^2)
\]

Distribute the negative sign across the second group:

\[
6 + 6a^2 + 6h^2 - 12ah - 6 - a^2
\]

Now, combine like terms:
- The \( 6 \) and \( -6 \) cancel each other out.
- Combine \( 6a^2 \) and \( -a^2 \), which gives \( 5a^2 \).

Thus, the numerator becomes:

\[
5a^2 + 6h^2 - 12ah
\]

### Step 2: Divide by \( h \)

Now, divide the simplified numerator by \( h \):

\[
\frac{5a^2 + 6h^2 - 12ah}{h}
\]

We can now distribute the \( h \) to each term:

\[
\frac{5a^2}{h} + \frac{6h^2}{h} - \frac{12ah}{h}
\]

This simplifies further to:

\[
\frac{5a^2}{h} + 6h - 12a
\]

### Final Result:

\[
\frac{5a^2}{h} + 6h - 12a
\]

Would you like to explore the details or have any questions on this? Here are 5 related questions to consider:

1. How does this expression behave as \( h \to 0 \)?
2. How would this expression change if we set \( a = 1 \)?
3. Could we apply this simplification process to a different function involving \( h \)?
4. How can you use limits to further simplify expressions like this?
5. What are the practical applications of simplifying such expressions?

**Tip:** Always cancel common terms in the numerator and denominator to simplify expressions further!

Learn how to simplify the algebraic expression ((6 + 6a^2 + 6h^2 = 12ah) - (6 + a^2))/h. This problem involves algebraic simplification and division of polynomials, providing a step-by-step breakdown. Suitable for students in Grades 9-11.

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