36( x - 2 )³/ 32 - 16x

To solve the expression \( \frac{36( x - 2 )^3}{32} - 16x \), let's follow these steps:

1. Simplify the fraction: We start with the term \( \frac{36( x - 2 )^3}{32} \):
- We can simplify this fraction. First, notice that both 36 and 32 share a common factor: 4.
- Divide 36 and 32 by 4.
- \( \frac{36}{4} = 9 \)
- \( \frac{32}{4} = 8 \)
- So, we rewrite the term as:
- \( \frac{36( x - 2 )^3}{32} = \frac{9( x - 2 )^3}{8} \)

2. Keep the second term: The expression now looks like:
- \( \frac{9( x - 2 )^3}{8} - 16x \)

3. Find a common denominator: To combine these two terms, we can express \( 16x \) with the same denominator of 8:
- Since \( 16x = \frac{16x \cdot 8}{8} = \frac{128x}{8} \)

4. Combine the terms: Now we combine the fractions:
- \( \frac{9( x - 2 )^3}{8} - \frac{128x}{8} = \frac{9( x - 2 )^3 - 128x}{8} \)

5. Expand the numerator: Now, we expand \( ( x - 2 )^3 \):
- Using the binomial theorem or direct multiplication:
- \( (x - 2)(x - 2)(x - 2) = (x^2 - 4x + 4)(x - 2) = x^3 - 6x^2 + 12x - 8 \)
- So, substituting this back, we have:
- \( 9( x - 2 )^3 = 9(x^3 - 6x^2 + 12x - 8) = 9x^3 - 54x^2 + 108x - 72 \)

6. Put everything together: Now substitute this back in:
- \( \frac{9x^3 - 54x^2 + 108x - 72 - 128x}{8} = \frac{9x^3 - 54x^2 - 20x - 72}{8} \)

The final result of the expression is:

Answer: (9x^3 - 54x^2 - 20x - 72) / 8

This expression indicates that we have performed all the steps of simplifying, expanding, and combining like terms correctly. This result is in the simplified form of the original expression that we started with.