For all `x, (x^(2)-3x+1)(x+2)=?`

Options

A. `x^(3)-x^(2)-5x+2`
B. `x^(3)-x^(2)-5x-2`
C. `x^(3)-x^(2)+5x+2`
D. `x^(3)+x^(2)-5x+2`

Answer

To solve the expression \((x^2 - 3x + 1)(x + 2)\), we will use the distributive property (also known as the FOIL method for binomials). Here’s a step-by-step breakdown:

### Step 1: Distribute \(x + 2\) to each term in \(x^2 - 3x + 1\)

We will multiply each term in the first polynomial by each term in the second polynomial:

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

### Step 2: Perform the multiplications

Now we will calculate each multiplication:

1. \(x^2 \cdot x = x^3\)
2. \(x^2 \cdot 2 = 2x^2\)
3. \(-3x \cdot x = -3x^2\)
4. \(-3x \cdot 2 = -6x\)
5. \(1 \cdot x = x\)
6. \(1 \cdot 2 = 2\)

Putting these together, we have:

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

### Step 3: Combine like terms

Now we will combine the like terms:

1. For \(x^2\) terms: \(2x^2 - 3x^2 = -x^2\)
2. For \(x\) terms: \(-6x + x = -5x\)

So, combining everything gives:

\[ x^3 - x^2 - 5x + 2 \]

### Final Result

Thus, the expression \((x^2 - 3x + 1)(x + 2)\) simplifies to:

\[ \boxed{x^3 - x^2 - 5x + 2} \]

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