Express the area of the entire rectangle.

Question

Your answer should be a polynomial in standard form.
An area model for a rectangle that has a height of x plus eight and a width of x plus one. The rectangle is broken into four rectangles to isolate each term in the height and the width. The top left rectangle has a height of x and a width of x. The top right rectangle has a height of x and width of one. The bottom left rectangle has a height of eight and a width of x. The bottom right rectangle has a height of eight and a width of one.
\[x\]

Answer 1

To find the area of the rectangle, we need to multiply the height and the width. The height of the rectangle is given as \( (x + 8) \) and the width is \( (x + 1) \).

The area \( A \) is calculated as follows:

\[ A = \text{Height} \times \text{Width} = (x + 8)(x + 1) \]

Now, we can expand this expression using the distributive property (also known as the FOIL method for binomials):

\[ A = x(x + 1) + 8(x + 1) \]

Calculating each term:

1. \( x(x + 1) = x^2 + x \)
2. \( 8(x + 1) = 8x + 8 \)

Now, we can combine these results:

\[ A = x^2 + x + 8x + 8 \]

Combining like terms (the \( x \) terms):

\[ A = x^2 + 9x + 8 \]

Now, we have the area expressed as a polynomial in standard form:

\[ \boxed{x^2 + 9x + 8} \]