To determine the area of a rectangle given its length and width, we use the formula for the area of a rectangle, which is:
[tex]\[ \text{Area} = \text{Length} \times \text{Width} \][/tex]
In this case, we have the length of the rectangle as [tex]\( x + 1 \)[/tex] and the width as [tex]\( x + 11 \)[/tex].
### Step-by-Step Solution:
1. Write down the expression for the area of the rectangle:
[tex]\[ \text{Area} = (x + 1) \times (x + 11) \][/tex]
2. Expand the expression by using the distributive property (also known as FOIL method for binomials):
[tex]\[ (x + 1)(x + 11) = x \cdot x + x \cdot 11 + 1 \cdot x + 1 \cdot 11 \][/tex]
3. Calculate each part:
[tex]\[ x \cdot x = x^2 \][/tex]
[tex]\[ x \cdot 11 = 11x \][/tex]
[tex]\[ 1 \cdot x = x \][/tex]
[tex]\[ 1 \cdot 11 = 11 \][/tex]
4. Combine all the terms:
[tex]\[ x^2 + 11x + x + 11 \][/tex]
5. Simplify by combining like terms:
[tex]\[ x^2 + 11x + x + 11 = x^2 + 12x + 11 \][/tex]
Therefore, the area of the rectangle is given by the expression:
[tex]\[ x^2 + 12x + 11 \][/tex]
Given the options provided, the correct answer is:
[tex]\[ x^2 + 12x + 11 \][/tex]
So, the correct representation of the area of this rectangle is:
[tex]\[ x^2 + 12x + 11 \][/tex]
