In the product `(x^(2) -2) (1-3x+2x^(2))` the sum of coefficients of `x^(2) and x` is

Options

A. 5
B. 3
C. 6
D. 2

Answer

To solve the problem, we need to find the sum of the coefficients of \(x^2\) and \(x\) in the product \((x^2 - 2)(1 - 3x + 2x^2)\).

### Step 1: Expand the expression
We start by expanding the product \((x^2 - 2)(1 - 3x + 2x^2)\).

\[ (x^2 - 2)(1 - 3x + 2x^2) = x^2 \cdot (1 - 3x + 2x^2) - 2 \cdot (1 - 3x + 2x^2) \]

### Step 2: Distribute \(x^2\)
Now we distribute \(x^2\) across each term in the second polynomial:

\[ x^2 \cdot 1 = x^2 \]
\[ x^2 \cdot (-3x) = -3x^3 \]
\[ x^2 \cdot (2x^2) = 2x^4 \]

So, the result from this part is:

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

### Step 3: Distribute \(-2\)
Next, we distribute \(-2\) across each term in the second polynomial:

\[ -2 \cdot 1 = -2 \]
\[ -2 \cdot (-3x) = 6x \]
\[ -2 \cdot (2x^2) = -4x^2 \]

So, the result from this part is:

\[ -2 + 6x - 4x^2 \]

### Step 4: Combine the results
Now we combine the results from Step 2 and Step 3:

\[ (x^2 - 3x^3 + 2x^4) + (-2 + 6x - 4x^2) = 2x^4 - 3x^3 + (x^2 - 4x^2) + 6x - 2 \]

This simplifies to:

\[ 2x^4 - 3x^3 - 3x^2 + 6x - 2 \]

### Step 5: Identify the coefficients
Now we identify the coefficients of \(x^2\) and \(x\):

- The coefficient of \(x^2\) is \(-3\).
- The coefficient of \(x\) is \(6\).

### Step 6: Calculate the sum of the coefficients
Finally, we calculate the sum of the coefficients of \(x^2\) and \(x\):

\[ -3 + 6 = 3 \]

### Conclusion
Thus, the sum of the coefficients of \(x^2\) and \(x\) is \(3\).

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