Simplify using following identiy:
`(a +- b) (a^(2) ab + b^(2)) = a^(3) +- b^(3)`
`(3x- (5)/(x)) (9x^(2) + 15+ (25)/(x^(2)))`

Answer

To simplify the expression \((3x - \frac{5}{x})(9x^2 + 15 + \frac{25}{x^2})\) using the identity \((a - b)(a^2 + ab + b^2) = a^3 - b^3\), we will follow these steps:

### Step 1: Identify \(a\) and \(b\)
In our case, we can identify:
- \(a = 3x\)
- \(b = \frac{5}{x}\)

### Step 2: Verify the second term
Now, we need to check if the second term \(9x^2 + 15 + \frac{25}{x^2}\) matches the form \(a^2 + ab + b^2\):
- Calculate \(a^2\):
\[ a^2 = (3x)^2 = 9x^2 \]
- Calculate \(ab\):
\[ ab = (3x) \left(\frac{5}{x}\right) = 15 \]
- Calculate \(b^2\):
\[ b^2 = \left(\frac{5}{x}\right)^2 = \frac{25}{x^2} \]

Now, we can see that:
\[ a^2 + ab + b^2 = 9x^2 + 15 + \frac{25}{x^2} \]
This confirms that the second term matches the required form.

### Step 3: Apply the identity
Now that we have confirmed the identity, we can apply it:
\[ (3x - \frac{5}{x})(9x^2 + 15 + \frac{25}{x^2}) = (3x)^3 - \left(\frac{5}{x}\right)^3 \]

### Step 4: Calculate \(a^3\) and \(b^3\)
- Calculate \(a^3\):
\[ (3x)^3 = 27x^3 \]
- Calculate \(b^3\):
\[ \left(\frac{5}{x}\right)^3 = \frac{125}{x^3} \]

### Step 5: Write the final expression
Substituting back into the equation gives us:
\[ (3x - \frac{5}{x})(9x^2 + 15 + \frac{25}{x^2}) = 27x^3 - \frac{125}{x^3} \]

### Final Answer
Thus, the simplified expression is:
\[ 27x^3 - \frac{125}{x^3} \]

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