The given mathematical expression involves simplifying a product of two rational expressions:

$\frac{4x^2 - 24x}{4x^4 + 12x^3 - 16x^2} \cdot \frac{x^2 - 9x + 8}{x^2 - 14x + 48}.$

Let's simplify it step by step.

Step 1: Factorize all the terms

1. Numerator of the first fraction: $4x^2 - 24x$ Factor out the greatest common factor (GCF): $4x^2 - 24x = 4x(x - 6).$

2. Denominator of the first fraction: $4x^4 + 12x^3 - 16x^2$ Factor out the GCF: $4x^4 + 12x^3 - 16x^2 = 4x^2(x^2 + 3x - 4).$ Now factorize $x^2 + 3x - 4$: $x^2 + 3x - 4 = (x + 4)(x - 1).$ So: $4x^4 + 12x^3 - 16x^2 = 4x^2(x + 4)(x - 1).$

3. Numerator of the second fraction: $x^2 - 9x + 8$ Factorize $x^2 - 9x + 8$: $x^2 - 9x + 8 = (x - 8)(x - 1).$

4. Denominator of the second fraction: $x^2 - 14x + 48$ Factorize $x^2 - 14x + 48$: $x^2 - 14x + 48 = (x - 8)(x - 6).$

Step 2: Rewrite the expression with factors

Now rewrite the entire expression: $\frac{4x(x - 6)}{4x^2(x + 4)(x - 1)} \cdot \frac{(x - 8)(x - 1)}{(x - 8)(x - 6)}.$

Step 3: Simplify the expression

Cancel out common factors from the numerator and denominator:

1. $4x$ in the numerator and $4x^2$ in the denominator cancel partially, leaving $\frac{1}{x}$.
2. $(x - 6)$ cancels.
3. $(x - 8)$ cancels.
4. $(x - 1)$ cancels.

After canceling, we are left with: $\frac{1}{x(x + 4)}.$

Final Answer:

The simplified expression is: $\frac{1}{x(x + 4)}.$

Would you like a more detailed explanation or help with a similar problem? Here are 5 related questions to expand on this:

1. How do you factorize quadratic expressions like $x^2 - 9x + 8$?
2. What is the process of canceling terms in rational expressions?
3. How do you find the domain of a rational expression like $\frac{1}{x(x + 4)}$?
4. Can you simplify other products of rational expressions with higher powers?
5. What happens if a factor is canceled but the variable is restricted (e.g., $x - 6 = 0$)?

Tip: Always check for restrictions on the domain (values that make the denominator zero) after simplifying a rational expression.