Simplify :`(8x^(2)-12x^(3)+6x)/(2x)`

Explanation

To simplify the expression \((8x^{2}-12x^{3}+6x)/(2x)\), we can follow these steps:

### Step 1: Separate the terms in the numerator
We can rewrite the expression by separating each term in the numerator over the denominator:
\[ \frac{8x^{2}}{2x} - \frac{12x^{3}}{2x} + \frac{6x}{2x} \]

### Step 2: Simplify each term
Now, we simplify each term individually:
1. For \(\frac{8x^{2}}{2x}\):
\[ \frac{8}{2} \cdot \frac{x^{2}}{x} = 4x \]

2. For \(\frac{12x^{3}}{2x}\):
\[ \frac{12}{2} \cdot \frac{x^{3}}{x} = 6x^{2} \]

3. For \(\frac{6x}{2x}\):
\[ \frac{6}{2} \cdot \frac{x}{x} = 3 \]

### Step 3: Combine the simplified terms
Now we can combine the simplified terms:
\[ 4x - 6x^{2} + 3 \]

### Step 4: Rearrange the expression
Rearranging the expression in standard polynomial form (highest degree first):
\[ -6x^{2} + 4x + 3 \]

### Final Answer
Thus, the simplified form of the expression \((8x^{2}-12x^{3}+6x)/(2x)\) is:
\[ -6x^{2} + 4x + 3 \]

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