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 expression \((x^2 - 2)(1 - 3x + 2x^2)\). Let's go through the steps:

### Step 1: Expand the expression
We will expand the product \((x^2 - 2)(1 - 3x + 2x^2)\) using the distributive property (also known as the FOIL method for binomials).

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

### Step 2: Distribute \(x^2\)
Now, we will distribute \(x^2\) across the terms in the second polynomial:

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

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

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

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

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

### Step 5: Combine like terms
Now we will combine like terms:

- The \(x^4\) term: \(2x^4\)
- The \(x^3\) term: \(-3x^3\)
- The \(x^2\) terms: \(x^2 - 4x^2 = -3x^2\)
- The \(x\) term: \(6x\)
- The constant term: \(-2\)

So, the expanded expression is:

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

### Step 6: Identify coefficients
Now we identify the coefficients of \(x^2\) and \(x\):

- Coefficient of \(x^2\) is \(-3\)
- Coefficient of \(x\) is \(6\)

### Step 7: Calculate the sum of the coefficients
Finally, we calculate the sum of the coefficients of \(x^2\) and \(x\):

\[ -3 + 6 = 3 \]

### Final Answer
The sum of the coefficients of \(x^2\) and \(x\) is \(3\).

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