To simplify the given expression, we'll break it down step by step.

Expression:

$\frac{x^5 + 9x^4 - 10x^3}{9x(x - 1)^2} \cdot \frac{6x^2 - 36x - 240}{x^2 - 100}$

Step 1: Factorize where possible

1. Factorize the numerator of the first fraction:

   • $x^5 + 9x^4 - 10x^3$ can be factored by taking out the common factor $x^3$: $x^5 + 9x^4 - 10x^3 = x^3(x^2 + 9x - 10)$
   • Now, we can factor $x^2 + 9x - 10$ further by finding factors of $-10$ that add up to $9$: $x^2 + 9x - 10 = (x + 10)(x - 1)$
   • Thus, the numerator becomes: $x^3(x + 10)(x - 1)$

2. Factorize the denominator of the second fraction:

   • $x^2 - 100$ is a difference of squares: $x^2 - 100 = (x + 10)(x - 10)$

3. Factorize the numerator of the second fraction:

   • $6x^2 - 36x - 240$ can be factored by taking out the common factor 6: $6x^2 - 36x - 240 = 6(x^2 - 6x - 40)$
   • Then factor $x^2 - 6x - 40$ by finding factors of $-40$ that add up to $-6$: $x^2 - 6x - 40 = (x - 10)(x + 4)$
   • Thus, the numerator becomes: $6(x - 10)(x + 4)$

Step 2: Substitute factorizations into the expression

$\frac{x^3(x + 10)(x - 1)}{9x(x - 1)^2} \cdot \frac{6(x - 10)(x + 4)}{(x + 10)(x - 10)}$

Step 3: Simplify the expression

1. Cancel out common factors:
   • $x$ cancels with $x$ in the denominator.
   • $x + 10$ cancels with $x + 10$.
   • $x - 1$ cancels one of the $x - 1$ in the denominator.
   • $x - 10$ cancels with $x - 10$.

This leaves us with: $\frac{x^2 \cdot 6(x + 4)}{9(x - 1)}$

2. Simplify the remaining expression:
   • Distribute the $6$ in the numerator: $6x^2(x + 4) = 6x^3 + 24x^2$

Final simplified expression: $\frac{6x^3 + 24x^2}{9(x - 1)}$

3. Further simplification (if possible):
   • Factor out the common factor $3$ from the numerator: $\frac{6x^3 + 24x^2}{9(x - 1)} = \frac{3(2x^3 + 8x^2)}{9(x - 1)} = \frac{2x^3 + 8x^2}{3(x - 1)}$

Thus, the fully simplified expression is: $\boxed{\frac{2x^3 + 8x^2}{3(x - 1)}}$

Would you like details on any specific step, or have any further questions?

---

Relative Questions:

1. How do you factor polynomials like $x^2 + 9x - 10$?
2. What are the rules for simplifying expressions with exponents?
3. Can you explain the difference of squares formula?
4. How do you handle fractions with polynomial expressions in the numerator and denominator?
5. How do you determine if further simplification is possible?

Tip: Always look for common factors in both the numerator and denominator to simplify expressions effectively.