Factorization of Polynomial or expression-
`x^(2) + 5x + 6`

Answer

To factor the polynomial \( x^2 + 5x + 6 \), we can follow these steps:

### Step 1: Identify the coefficients
The given polynomial is in the standard form \( ax^2 + bx + c \), where:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = 5 \) (coefficient of \( x \))
- \( c = 6 \) (constant term)

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( ac = 1 \times 6 = 6 \) and add up to \( b = 5 \).

The pairs of factors of 6 are:
- \( 1 \times 6 \)
- \( 2 \times 3 \)

Among these, the pair \( 2 \) and \( 3 \) adds up to \( 5 \).

### Step 3: Rewrite the middle term using the two numbers
We can rewrite the polynomial by splitting the middle term \( 5x \) into \( 2x + 3x \):
\[ x^2 + 2x + 3x + 6 \]

### Step 4: Group the terms
Now, we can group the terms:
\[ (x^2 + 2x) + (3x + 6) \]

### Step 5: Factor out the common factors from each group
From the first group \( (x^2 + 2x) \), we can factor out \( x \):
\[ x(x + 2) \]
From the second group \( (3x + 6) \), we can factor out \( 3 \):
\[ 3(x + 2) \]

### Step 6: Combine the factored terms
Now, we can combine the factored terms:
\[ x(x + 2) + 3(x + 2) \]
Notice that \( (x + 2) \) is common in both terms, so we can factor it out:
\[ (x + 2)(x + 3) \]

### Conclusion
Thus, the factorization of the polynomial \( x^2 + 5x + 6 \) is:
\[ (x + 2)(x + 3) \]

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