In the expansion of `(2x^(2)-8)(x-4)^(2)`, find the value of :
`(i)` coefficient of `x^(3)` (ii) Coefficient of `x^(2)`
(iii) Constant term

Answer

To solve the problem, we need to expand the expression \((2x^2 - 8)(x - 4)^2\) and then find the coefficients of \(x^3\), \(x^2\), and the constant term.

### Step 1: Expand \((x - 4)^2\)

Using the formula for the square of a binomial, we have:
\[ (x - 4)^2 = x^2 - 2 \cdot 4 \cdot x + 4^2 = x^2 - 8x + 16 \]

**Hint:** Remember the formula \((a - b)^2 = a^2 - 2ab + b^2\).

### Step 2: Multiply \((2x^2 - 8)\) with the expanded form of \((x - 4)^2\)

Now we will multiply \(2x^2 - 8\) with \(x^2 - 8x + 16\):
\[ (2x^2 - 8)(x^2 - 8x + 16) \]

Distributing \(2x^2\):
\[ 2x^2 \cdot x^2 = 2x^4 \]
\[ 2x^2 \cdot (-8x) = -16x^3 \]
\[ 2x^2 \cdot 16 = 32x^2 \]

Distributing \(-8\):
\[ -8 \cdot x^2 = -8x^2 \]
\[ -8 \cdot (-8x) = 64x \]
\[ -8 \cdot 16 = -128 \]

### Step 3: 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 4: Identify the coefficients and constant term

From the final expression \(2x^4 - 16x^3 + 24x^2 + 64x - 128\):
- The coefficient of \(x^3\) is \(-16\).
- The coefficient of \(x^2\) is \(24\).
- The constant term is \(-128\).

### Final Answers:
(i) Coefficient of \(x^3\) = \(-16\)
(ii) Coefficient of \(x^2\) = \(24\)
(iii) Constant term = \(-128\)

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