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To solve and simplify the given expression, we can proceed step by step. The expression is:

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

Let's break it down:

### Step 1: Simplify Each Component
1. **First Fraction:**
   \[
   \frac{(a^2 - 3a)^2}{9 - a^2}
   \]
   The numerator is \((a^2 - 3a)^2\). The denominator can be factored:
   \[
   9 - a^2 = (3 - a)(3 + a)
   \]
   So the fraction becomes:
   \[
   \frac{(a(a - 3))^2}{(3 - a)(3 + a)} = \frac{a^2(a - 3)^2}{(3 - a)(3 + a)}
   \]

   Note: \( (3 - a) = -(a - 3) \), so we have:
   \[
   \frac{a^2(a - 3)^2}{-(a - 3)(3 + a)} = -\frac{a^2(a - 3)}{(a + 3)}
   \]

2. **Second Fraction:**
   \[
   \frac{27 - a^3}{(a + 3)^2 - 3a}
   \]
   The numerator \( 27 - a^3 \) can be factored as:
   \[
   27 - a^3 = (3 - a)(9 + 3a + a^2)
   \]
   The denominator:
   \[
   (a + 3)^2 - 3a =

Learn how to solve and simplify complex algebraic fractions step-by-step. This problem involves fraction simplification and factoring techniques. Suitable for advanced high school students.

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