`6(2x-(3)/(x))^2+7(2x-(3)/(x))-20`

Answer

To factorize the polynomial \( 6\left(2x - \frac{3}{x}\right)^2 + 7\left(2x - \frac{3}{x}\right) - 20 \), we will follow these steps:

### Step 1: Substitute the expression
Let \( p = 2x - \frac{3}{x} \). Then, we can rewrite the polynomial as:
\[ 6p^2 + 7p - 20 \]

**Hint:** When dealing with complex expressions, substituting a simpler variable can make the problem easier to manage.

### Step 2: Factor the quadratic expression
We need to factor \( 6p^2 + 7p - 20 \). To do this, we look for two numbers that multiply to \( 6 \times (-20) = -120 \) and add to \( 7 \). The numbers that work are \( 15 \) and \( -8 \).

**Hint:** When factoring quadratics, find two numbers that multiply to the product of the leading coefficient and the constant term, and add to the middle coefficient.

### Step 3: Rewrite the middle term
Using these numbers, we can rewrite the quadratic:
\[ 6p^2 + 15p - 8p - 20 \]

**Hint:** Breaking down the middle term helps in grouping the polynomial for easier factorization.

### Step 4: Group the terms
Now, we group the terms:
\[ (6p^2 + 15p) + (-8p - 20) \]

**Hint:** Grouping allows us to factor by common terms in pairs.

### Step 5: Factor out the common terms
From the first group, we can factor out \( 3p \):
\[ 3p(2p + 5) - 4(2p + 5) \]
Now we can factor out \( (2p + 5) \):
\[ (2p + 5)(3p - 4) \]

**Hint:** Look for common factors in grouped terms to simplify the expression.

### Step 6: Substitute back the value of \( p \)
Now, we substitute back \( p = 2x - \frac{3}{x} \):
\[ (2(2x - \frac{3}{x}) + 5)(3(2x - \frac{3}{x}) - 4) \]

**Hint:** Always remember to substitute back to the original variable after factorization.

### Step 7: Simplify the factors
Now, simplify each factor:
1. First factor:
\[ 2(2x - \frac{3}{x}) + 5 = 4x - \frac{6}{x} + 5 = 4x + 5 - \frac{6}{x} \]
2. Second factor:
\[ 3(2x - \frac{3}{x}) - 4 = 6x - \frac{9}{x} - 4 = 6x - 4 - \frac{9}{x} \]

**Hint:** Simplifying each factor can help in understanding the final expression better.

### Final Answer
Thus, the factorized form of the polynomial is:
\[ (2(2x - \frac{3}{x}) + 5)(3(2x - \frac{3}{x}) - 4) \]

This can be further simplified if needed, but the main factorization is complete.