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

Answer

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

### Step 1: Expand \( (x - 4)^2 \)
Using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \), we can expand \( (x - 4)^2 \).

\[ (x - 4)^2 = x^2 - 2 \cdot 4 \cdot x + 4^2 = x^2 - 8x + 16 \]

### Step 2: Substitute the expansion back into the expression
Now 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 each term in \( (2x^2 - 8) \):

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

Calculating each part:

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

So, the first part gives us:

\[ 2x^4 - 16x^3 + 32x^2 \]

Now for the second part:

1. \( -8 \cdot x^2 = -8x^2 \)
2. \( -8 \cdot (-8x) = 64x \)
3. \( -8 \cdot 16 = -128 \)

So, the second part gives us:

\[ -8x^2 + 64x - 128 \]

### Step 4: Combine like terms
Now, we combine the results from both parts:

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

This simplifies to:

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

### Step 5: Identify the constant term
The constant term in the expression is the term that does not contain \( x \). From our final expression, the constant term is:

\[ -128 \]

### Final Answer
The value of the constant term is \( -128 \).

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