Factor the following.
`x^(2)+22xy+121y^(2)=0`

Answer

To factor the quadratic equation \( x^2 + 22xy + 121y^2 = 0 \), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is in the form \( ax^2 + bxy + cy^2 = 0 \), where:
- \( a = 1 \)
- \( b = 22 \)
- \( c = 121 \)

### Step 2: Factor the middle term
We need to express the middle term \( 22xy \) as a sum of two terms whose product equals \( ac = 1 \cdot 121 = 121 \) and whose sum equals \( 22 \).

We can split \( 22xy \) into \( 11xy + 11xy \).

### Step 3: Rewrite the equation
Now we can rewrite the equation as:
\[ x^2 + 11xy + 11xy + 121y^2 = 0 \]

### Step 4: Group the terms
Next, we can group the terms:
\[ (x^2 + 11xy) + (11xy + 121y^2) = 0 \]

### Step 5: Factor by grouping
Now we factor out the common terms:
\[ x(x + 11y) + 11y(x + 11y) = 0 \]

### Step 6: Factor out the common binomial
Now we can factor out the common binomial \( (x + 11y) \):
\[ (x + 11y)(x + 11y) = 0 \]

### Step 7: Write the final factored form
This can be written as:
\[ (x + 11y)^2 = 0 \]

### Conclusion
Thus, the factored form of the quadratic equation \( x^2 + 22xy + 121y^2 = 0 \) is:
\[ (x + 11y)^2 = 0 \]