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Let's analyze the given expression and simplify it step by step.

The expression is:

\[
\frac{(a^2 - 3a)^2}{9 - a^2} \times \frac{27 - a^3}{(a + 3)^2 - 3a} \div \frac{a^4 - 9a^2}{(a^2 + 3a)^2}
\]

### Step 1: Simplify Each Fraction Individually

1. **First Fraction:**

\[
\frac{(a^2 - 3a)^2}{9 - a^2}
\]

Notice that \( a^2 - 3a = a(a - 3) \), so:

\[
\frac{(a(a - 3))^2}{9 - a^2} = \frac{a^2(a - 3)^2}{9 - a^2}
\]

The denominator \( 9 - a^2 \) can be factored as:

\[
9 - a^2 = (3 - a)(3 + a)
\]

So the fraction becomes:

\[
\frac{a^2(a - 3)^2}{(3 - a)(3 + a)}
\]

Since \(3 - a = -(a - 3)\), we can rewrite this as:

\[
\frac{a^2(a - 3)^2}{-(a - 3)(a + 3)} = \frac{-a^2(a - 3)}{a + 3}
\]

2. **Second Fraction:**

\[
\frac{27 - a^3}{(a + 3)^2 - 3a}
\]

Recognize that \( 27 - a^3 \) is a difference of cubes:

\[
27 - a^3 = (3 - a)(9 + 3a + a^2)
\]

The denominator \( (a + 3)^2 - 3a \) needs expansion:

\[
(a + 3)^2 = a^2 + 6a + 9
\]

So:

\[
(a + 3)^2 - 3a = a^2 + 6a + 9 - 3a = a^2 + 3a + 9
\]

Thus, the second fraction simplifies to:

\[
\frac{(3 - a)(9 + 3a + a^2)}{a^2 + 3a + 9}
\]

This simplifies further to:

\[
3 - a
\]

3. **Third Fraction:**

\[
\frac{a^4 - 9a^2}{(a^2 + 3a)^2}
\]

Factor the numerator \( a^4 - 9a^2 \):

\[
a^4 - 9a^2 = a^2(a^2 - 9) = a^2(a - 3)(a + 3)
\]

The denominator can be expanded:

\[
(a^2 + 3a)^2 = (a(a + 3))^2 = a^2(a + 3)^2
\]

Thus, the third fraction simplifies to:

\[
\frac{a^2(a - 3)(a + 3)}{a^2(a + 3)^2} = \frac{a - 3}{a + 3}
\]

### Step 2: Combine All Fractions

Now, substitute back into the original expression:

\[
\frac{-a^2(a - 3)}{a + 3} \times (3 - a) \div \frac{a - 3}{a + 3}
\]

Multiply and simplify:

The division by a fraction is equivalent to multiplying by its reciprocal:

\[
\frac{-a^2(a - 3)}{a + 3} \times (3 - a) \times \frac{a + 3}{a - 3}
\]

This cancels out the \( a + 3 \) and \( a - 3 \) terms, so the simplified result is:

\[
-a^2(3 - a)
\]

Simplify further:

\[
-a^2(3 - a) = -3a^2 + a^3 = a^3 - 3a^2
\]

### Final Simplified Expression:

\[
a^3 - 3a^2
\]

Would you like to go through the steps in more detail or explore further? Here are some related questions:

1. How does the difference of squares formula work?
2. What is the difference between factoring and expanding polynomials?
3. How can you identify when to factor an expression?
4. What is the importance of simplifying algebraic expressions?
5. How do you handle the division of algebraic fractions?

**Tip:** Always check for common factors in the numerator and denominator before simplifying any algebraic fraction.

Learn how to simplify the algebraic expression (a^3 - 3a^2) step-by-step with detailed explanations. This problem involves factorization, fraction simplification, and algebraic manipulation. Suitable for students in Grades 10-12.

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