Calculate Square Area: Finding Area When x+1 Side Length Transforms to Rectangle

First, let's recall the formulas for calculating square area:

The area of a square (where all sides are equal and all angles are $90\degree$) with a side length of $a$(length units - u)

, is given by the formula:

$\boxed{ S_{\textcolor{red}{\boxed{}}}=a^2}$(square units - sq.u),

After recalling this fact, let's solve the problem:

First, let's mark the square's vertices with letters: $ABCD$

Next, considering the given data (that the square's side length is: $x+1$ cm), we'll use the above square area formula to express the area of the given square using its side length-$AB=BC=CD=DA= x +1$ (cm):

$S_{\textcolor{red}{\boxed{}}}=AB^2\\ \downarrow\\ S_{\textcolor{red}{\boxed{}}}=(x+1)^2$(sq.cm)

We'll continue and simplify the algebraic expression we got for the square's area, this will be done using the shortened multiplication formula for squaring a binomial:

$(c+d)^2=c^2+2cd+d^2$Therefore, we'll apply this formula to our square area expression:

$S_{\textcolor{red}{\boxed{}}}=(x+1)^2 \\ \downarrow\\ \boxed{S_{\textcolor{red}{\boxed{}}}=x^2+2x+1}$(sq.cm)

Therefore, the correct answer is answer D.