Difference of squares: Worded problems

First, recall the formula for calculating the area of a rectangle:

The area of a rectangle (which has two pairs of equal opposite sides and all angles are $90\degree$) with sides of length $a,\hspace{4pt} b$ units, is given by the formula:

$\boxed{ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=a\cdot b }$(square units)

Proceed to solve the problem:

Calculate the area of the rectangle whose vertices we'll mark with letters $EFGH$ (drawing)

$$It is given in the problem that one side of the rectangle is obtained by extending one side of the square with side length $x +1$ (cm) by 1 cm, and the second side of the rectangle is obtained by shortening the adjacent side of the given square by 1 cm:

Therefore, the lengths of the rectangle's sides are:

$EF=HG=(x+1)+1\\ \downarrow\\ \boxed{ EF=HG=x+2}\\ \hspace{2pt}\\ \\ EH=FG=(x+1)-1\\ \downarrow\\ \boxed{ EH=FG=x }$ (cm)

Apply the above formula to calculate the area of the rectangle that was formed from the square in the way described in the problem:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=EF\cdot EH\\ \downarrow\\ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=(x+2)x$ (sq cm)

Continue to simplify the expression that we obtained for the rectangle's area, using the distributive property:

$(m+n)d=md+nd$ Therefore, applying the distributive property, we obtain the following area for the rectangle:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=(x+2)x \\ \boxed{ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=x^2+2x}$(sq cm)

The correct answer is answer B.

First, let's recall the formula for calculating the area of a rectangle:

The area of a rectangle (which has two pairs of equal opposite sides and all angles are $90\degree$) with sides of length $a,\hspace{4pt} b$units, is given by the formula:

$\boxed{ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=a\cdot b }$(square units)

With this is mind, let's proceed to solve the problem:

Calculate the area of the rectangle whose vertices we'll mark with letters $EFGH$

$$We are told that one side of the rectangle is obtained by extending one side of the square with side length $x$ (cm) by 3 cm, and the second side of the rectangle is obtained by shortening the adjacent side of the given square by 3 cm:

$$

Therefore, the lengths of the rectangle's sides are:

$EF=HG=x+3\\ EH=FG=x-3$cm,

Apply the above formula in order to calculate the area of the rectangle that was formed from the square as described in the question:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=EF\cdot EH\\ \downarrow\\ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=(x+3)(x-3)$ (sq cm)

Continue to simplify the expression that we obtained for the rectangle's area, using the difference of squares formula:

$(c+d)(c-d)=c^2-d^2$The area of the rectangle using the above formula is as follows:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=(x+3)(x-3) \\ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=x^2-3^2\\ \boxed{ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=x^2-9}$(sq cm)

Therefore, the correct answer is answer C.