Let's fully simplify the given expression step-by-step:

$\left( \frac{4x^3 + 56x^2 + 160x}{x + 10} \right) \cdot \left( \frac{81 - x^2}{-x - 9} \right) \cdot \left( \frac{3x}{x^2 - 5x - 36} \right)$

Step 1: Factor each expression.

First fraction:

$\frac{4x^3 + 56x^2 + 160x}{x + 10}$ Factor the numerator: $4x(x^2 + 14x + 40)$ We now factor the quadratic $x^2 + 14x + 40$. Using the quadratic formula or trial and error, this quadratic doesn't factor neatly, so we'll leave it as is for now: $\frac{4x(x^2 + 14x + 40)}{x + 10}$

Second fraction:

$\frac{81 - x^2}{-x - 9}$ We recognize that $81 - x^2$ is a difference of squares: $81 - x^2 = (9 - x)(9 + x)$ Thus, we rewrite the fraction: $\frac{(9 - x)(9 + x)}{-(x + 9)} = \frac{(9 - x)(9 + x)}{-(9 + x)}$ The $9 + x$ cancels out, leaving: $\frac{9 - x}{-1} = -(9 - x)$ This simplifies to: $x - 9$

Third fraction:

$\frac{3x}{x^2 - 5x - 36}$ We factor the quadratic $x^2 - 5x - 36$: $x^2 - 5x - 36 = (x - 9)(x + 4)$ Thus, the fraction becomes: $\frac{3x}{(x - 9)(x + 4)}$

Step 2: Combine the expressions.

We now have: $\left( \frac{4x(x^2 + 14x + 40)}{x + 10} \right) \cdot (x - 9) \cdot \left( \frac{3x}{(x - 9)(x + 4)} \right)$

Step 3: Simplify.

• The $x - 9$ terms cancel out.
• Combine the remaining terms: $\frac{4x(x^2 + 14x + 40)}{x + 10} \cdot \frac{3x}{x + 4}$

Multiply the numerators and the denominators: $\frac{4x(3x)(x^2 + 14x + 40)}{(x + 10)(x + 4)}$ Simplifying the numerator: $12x^2(x^2 + 14x + 40)$ Thus, the expression becomes: $\frac{12x^2(x^2 + 14x + 40)}{(x + 10)(x + 4)}$

This is the fully simplified expression.

Summary:

$\frac{12x^2(x^2 + 14x + 40)}{(x + 10)(x + 4)}$

Would you like further details or clarifications on any step?

Related Questions:

1. How do you factor complex quadratic expressions?
2. What is the difference of squares and how is it used in simplification?
3. How do you cancel common factors in rational expressions?
4. What are the common factoring techniques for polynomials?
5. How does multiplying rational expressions work?

Tip:

Always check for common factors across numerators and denominators before multiplying or simplifying rational expressions.