Factorization of Polynomial or expression-
`x^(2) + 7xy + 12y^(2)`

Answer

To factor the polynomial \( x^2 + 7xy + 12y^2 \), we will follow these steps:

### Step 1: Identify the structure of the polynomial
The polynomial is a quadratic in terms of \( x \) with coefficients dependent on \( y \). It can be expressed in the standard form \( ax^2 + bxy + cy^2 \).

### Step 2: Rewrite the middle term
We need to find two numbers that multiply to \( ac \) (where \( a = 1 \) and \( c = 12y^2 \), so \( ac = 12y^2 \)) and add up to \( b \) (where \( b = 7y \)).

The two numbers that satisfy these conditions are \( 3y \) and \( 4y \) because:
- \( 3y \times 4y = 12y^2 \)
- \( 3y + 4y = 7y \)

### Step 3: Rewrite the polynomial
Now, we can rewrite the polynomial using these two numbers:
\[ x^2 + 3xy + 4xy + 12y^2 \]

### Step 4: Group the terms
Next, we group the terms:
\[ (x^2 + 3xy) + (4xy + 12y^2) \]

### Step 5: Factor out the common terms from each group
From the first group \( (x^2 + 3xy) \), we can factor out \( x \):
\[ x(x + 3y) \]
From the second group \( (4xy + 12y^2) \), we can factor out \( 4y \):
\[ 4y(x + 3y) \]

### Step 6: Combine the factored terms
Now, we can combine the factored terms:
\[ x(x + 3y) + 4y(x + 3y) \]
This can be factored further by taking out the common factor \( (x + 3y) \):
\[ (x + 3y)(x + 4y) \]

### Final Answer
Thus, the factorization of the polynomial \( x^2 + 7xy + 12y^2 \) is:
\[ (x + 3y)(x + 4y) \]

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