Factories:
`1 + 2x + x ^(2)`

Answer

To factor the expression \(1 + 2x + x^2\), we can follow these steps:

### Step 1: Rearrange the expression
We start by rearranging the expression in standard form:
\[ x^2 + 2x + 1 \]

### Step 2: Identify the structure
Notice that the expression resembles a perfect square trinomial. A perfect square trinomial takes the form:
\[ (a + b)^2 = a^2 + 2ab + b^2 \]
In our case, we can identify \(a = x\) and \(b = 1\).

### Step 3: Rewrite the expression
Using the perfect square trinomial structure, we can rewrite the expression as:
\[ (x + 1)^2 \]

### Step 4: Final factorization
Thus, the factorization of the expression \(1 + 2x + x^2\) is:
\[ (x + 1)(x + 1) \quad \text{or simply} \quad (x + 1)^2 \]

### Step 5: Set the factors to zero (optional)
If we want to find the roots of the equation \(1 + 2x + x^2 = 0\), we can set the factor equal to zero:
\[ x + 1 = 0 \]
This gives us:
\[ x = -1 \]

### Summary of the solution:
The factorization of \(1 + 2x + x^2\) is \((x + 1)^2\), and the value of \(x\) that satisfies the equation is \(x = -1\).

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