In the expansion of `(2x^(2)-8) (x-4)^(2)`, find the value of
coefficient of `x^(3)`

Answer

To find the coefficient of \( x^3 \) in the expansion of \( (2x^2 - 8)(x - 4)^2 \), we will follow these steps:

### Step 1: Expand \( (x - 4)^2 \)
Using the formula \( (x - y)^2 = x^2 - 2xy + y^2 \), we can expand \( (x - 4)^2 \):
\[ (x - 4)^2 = x^2 - 2(4)(x) + 4^2 = x^2 - 8x + 16 \]

### Step 2: Substitute the expansion back into the expression
Now we substitute \( (x - 4)^2 \) back into the original expression:
\[ (2x^2 - 8)(x^2 - 8x + 16) \]

### Step 3: Distribute \( (2x^2 - 8) \) across \( (x^2 - 8x + 16) \)
We will distribute \( 2x^2 - 8 \) across each term in \( x^2 - 8x + 16 \):
\[ = 2x^2 \cdot (x^2 - 8x + 16) - 8 \cdot (x^2 - 8x + 16) \]

Calculating each part:
1. \( 2x^2 \cdot x^2 = 2x^4 \)
2. \( 2x^2 \cdot (-8x) = -16x^3 \)
3. \( 2x^2 \cdot 16 = 32x^2 \)

Now for the second part:
1. \( -8 \cdot x^2 = -8x^2 \)
2. \( -8 \cdot (-8x) = 64x \)
3. \( -8 \cdot 16 = -128 \)

### Step 4: Combine all the terms
Now we combine all the terms:
\[ 2x^4 - 16x^3 + 32x^2 - 8x^2 + 64x - 128 \]

Combining like terms:
\[ 2x^4 - 16x^3 + (32x^2 - 8x^2) + 64x - 128 \]
\[ = 2x^4 - 16x^3 + 24x^2 + 64x - 128 \]

### Step 5: Identify the coefficient of \( x^3 \)
From the final expression \( 2x^4 - 16x^3 + 24x^2 + 64x - 128 \), we can see that the coefficient of \( x^3 \) is:
\[ \text{Coefficient of } x^3 = -16 \]

### Final Answer
The coefficient of \( x^3 \) in the expansion of \( (2x^2 - 8)(x - 4)^2 \) is \( -16 \).

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