Fully simplify the expression below and write your answer as a single fraction. \[ \frac{x^{4}-6 x^{3}}{4 x^{2}-20 x+24} \cdot \frac{2 x^{2}+6 x-36}{x^{2}-36} \]

To simplify the expression \[ \frac{x^{4}-6 x^{3}}{4 x^{2}-20 x+24} \cdot \frac{2 x^{2}+6 x-36}{x^{2}-36}, \] we first factor each polynomial. 1. **Numerator 1:** \(x^4 - 6x^3\) We can factor this as: \[ x^3(x - 6). \] 2. **Denominator 1:** \(4x^2 - 20x + 24\) We can factor this by looking for common factors: \[ 4(x^2 - 5x + 6) = 4(x - 2)(x - 3). \] 3. **Numerator 2:** \(2x^2 + 6x - 36\) Factor out the common factor: \[ 2(x^2 + 3x - 18) = 2(x - 3)(x + 6). \] 4. **Denominator 2:** \(x^2 - 36\) This is a difference of squares: \[ (x - 6)(x + 6). \] Now substituting the factors back into the expression, we get: \[ \frac{x^3(x - 6)}{4(x - 2)(x - 3)} \cdot \frac{2(x - 3)(x + 6)}{(x - 6)(x + 6)}. \] We can combine these fractions: \[ \frac{x^3(x - 6) \cdot 2(x - 3)(x + 6)}{4(x - 2)(x - 3)(x - 6)(x + 6)}. \] Next, we can cancel out common factors. Notably, we see \(x - 6\) in the numerator and denominator, and \(x - 3\) in the numerator and denominator: \[ = \frac{x^3 \cdot 2}{4(x - 2)(x + 6)}. \] After canceling, we can simplify \(2\) and \(4\): \[ = \frac{x^3}{2(x - 2)(x + 6)}. \] Thus, the fully simplified expression is: \[ \boxed{\frac{x^3}{2(x - 2)(x + 6)}}. \]