Expand: `(2x-(1)/(x)) (3x+(2)/(x))`

Answer

To expand the expression \((2x - \frac{1}{x})(3x + \frac{2}{x})\), we will use the distributive property (also known as the FOIL method for binomials). Here’s a step-by-step solution:

### Step 1: Write down the expression
We start with the expression:
\[ (2x - \frac{1}{x})(3x + \frac{2}{x}) \]

### Step 2: Distribute the first term
First, we multiply \(2x\) by each term in the second bracket:
\[ 2x \cdot 3x + 2x \cdot \frac{2}{x} \]
Calculating these:
1. \(2x \cdot 3x = 6x^2\)
2. \(2x \cdot \frac{2}{x} = 4\) (since \(x\) cancels out)

So, from this step, we get:
\[ 6x^2 + 4 \]

### Step 3: Distribute the second term
Next, we multiply \(-\frac{1}{x}\) by each term in the second bracket:
\[ -\frac{1}{x} \cdot 3x - \frac{1}{x} \cdot \frac{2}{x} \]
Calculating these:
1. \(-\frac{1}{x} \cdot 3x = -3\) (since \(x\) cancels out)
2. \(-\frac{1}{x} \cdot \frac{2}{x} = -\frac{2}{x^2}\)

So, from this step, we get:
\[ -3 - \frac{2}{x^2} \]

### Step 4: Combine all the results
Now, we combine all the terms we obtained from both distributions:
\[ 6x^2 + 4 - 3 - \frac{2}{x^2} \]

### Step 5: Simplify the expression
Now, we simplify the constant terms:
\[ 6x^2 + (4 - 3) - \frac{2}{x^2} = 6x^2 + 1 - \frac{2}{x^2} \]

### Final Answer
Thus, the expanded form of the expression \((2x - \frac{1}{x})(3x + \frac{2}{x})\) is:
\[ 6x^2 + 1 - \frac{2}{x^2} \]