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

Answer

To find the coefficient of \(x^2\) in the expansion of \((2x^2 - 8)(x - 4)^2\), we can 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 this expansion back into the original expression:

\[ (2x^2 - 8)(x^2 - 8x + 16) \]

### Step 3: Distribute \(2x^2 - 8\) across the expanded polynomial
We will distribute \(2x^2 - 8\) across each term in \((x^2 - 8x + 16)\):

1. Multiply \(2x^2\) by each term:
- \(2x^2 \cdot x^2 = 2x^4\)
- \(2x^2 \cdot (-8x) = -16x^3\)
- \(2x^2 \cdot 16 = 32x^2\)

2. Multiply \(-8\) by each term:
- \(-8 \cdot x^2 = -8x^2\)
- \(-8 \cdot (-8x) = 64x\)
- \(-8 \cdot 16 = -128\)

### Step 4: Combine all the terms
Now we combine all the terms we obtained:

\[ 2x^4 - 16x^3 + (32x^2 - 8x^2) + 64x - 128 \]

This simplifies to:

\[ 2x^4 - 16x^3 + 24x^2 + 64x - 128 \]

### Step 5: Identify the coefficient of \(x^2\)
From the combined expression, the coefficient of \(x^2\) is \(24\).

### Final Answer
The coefficient of \(x^2\) in the expansion of \((2x^2 - 8)(x - 4)^2\) is \(24\).

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