Question: The image shows a partial screenshot of a computer desktop, likely running Windows. On the left…

The image shows a partial screenshot of a computer desktop, likely running Windows. On the left side, there’s a taskbar with various application icons including the Windows logo, search icon, and several other app icons. At the bottom left, there are two buttons labeled ‘Clear all’ and ‘Still builder’. On the right side of the image, there’s a partial view of what appears to be a question or problem statement, with the text ‘x^2 - 12x^2 + 12x + 12’ visible, which seems to be part of a mathematical expression. The top right corner shows ‘Question 4, 6:2:35’ which might be indicating a question number and a time. The image also shows an HP logo in the bottom left corner, suggesting this is an HP device.

Solution

The problem asks to find the least common denominator (LCD) for the given expressions, which are denominators:

\[ x^2 - 121, \quad x^2 + 22x + 121 \]

First, factor each expression.

1. Factor \(x^2 - 121\).

   This is a difference of squares:

   \[ x^2 - 121 = (x - 11)(x + 11) \]

2. Factor \(x^2 + 22x + 121\).

   This is a perfect square trinomial:

   \[ x^2 + 22x + 121 = (x + 11)^2 \]

Now, identify the LCD.

• The factors are \((x - 11)\) and \((x + 11)\) from the first expression, and \((x + 11)^2\) from the second expression.
• The least common denominator must include each factor the greatest number of times it appears in any factorization.

The LCD is:

\[ (x - 11)(x + 11)^2 \]

Thus, the least common denominator is \((x - 11)(x + 11)^2\).