Let's solve the problem step by step:
Given:
[tex]\[
\frac{4 x^2 - 8 x}{10 x^3} \cdot \frac{15 x^2 - 5 x}{x-2}
\][/tex]
### Step 1: Simplify the first fraction
First, simplify the fraction [tex]\(\frac{4 x^2 - 8 x}{10 x^3}\)[/tex].
Factor out the common terms:
[tex]\[ 4 x^2 - 8 x = 4x(x - 2) \][/tex]
[tex]\[ 10 x^3 = 10x^3 \][/tex]
So the fraction becomes:
[tex]\[
\frac{4 x (x - 2)}{10 x^3} = \frac{4 (x - 2)}{10 x^2} = \frac{2 (x - 2)}{5 x^2}
\][/tex]
### Step 2: Simplify the second fraction
Next, simplify the fraction [tex]\(\frac{15 x^2 - 5 x}{x - 2}\)[/tex].
Factor out the common terms:
[tex]\[ 15 x^2 - 5 x = 5 x (3 x - 1) \][/tex]
So the fraction becomes:
[tex]\[
\frac{5 x (3 x - 1)}{x - 2}
\][/tex]
### Step 3: Multiply the simplified fractions
Now, multiply the simplified fractions:
[tex]\[
\frac{2 (x - 2)}{5 x^2} \cdot \frac{5 x (3 x - 1)}{x - 2}
\][/tex]
### Step 4: Cancel out common factors
Notice that [tex]\((x - 2)\)[/tex] appears in both the numerator and denominator, so it cancels out:
[tex]\[
\frac{2 (x - 2)}{5 x^2} \cdot \frac{5 x (3 x - 1)}{x - 2} = \frac{2}{5 x^2} \cdot \frac{5 x (3 x - 1)}{1}
\][/tex]
Simplify further:
[tex]\[
\frac{2 \cdot 5 x \cdot (3 x - 1)}{5 x^2} = \frac{10 x (3 x - 1)}{5 x^2} = \frac{2 (3 x - 1)}{x}
\][/tex]
### Step 5: Final simple form
Thus, the expression simplifies to:
[tex]\[
6 - \frac{2}{x}
\][/tex]
Therefore, the final simplified result is:
[tex]\[
\frac{2 (3 x - 1)}{x}
\][/tex]
This concludes the detailed step-by-step solution for simplifying the given expression.
