A manufacturer is cutting a square opening out of a rectangular metal panel. The rectangul… — Armed Services Vocational Aptitude Battery (ASVAB)

A manufacturer is cutting a square opening out of a rectangular metal panel. The rectangular panel has a length of $(4x - 1)$ inches and a width of $(2x + 3)$ inches. The square opening to be removed has a side length of $(x + 2)$ inches. Which of the following expressions represents the total remaining area of the metal panel after the square is removed?

$7x^2 + 6x - 7$ Answer
B
$7x^2 + 10x - 7$
C
$7x^2 + 14x + 1$
D
$7x^2 - 7$

Answer

The correct expression is

7x^2 + 6x - 7

The remaining area is found by subtracting the square's area from the rectangle's area. Multiplying the dimensions

(4x - 1)(2x + 3)

yields

8x^2 + 10x - 3

. Squaring the dimension

(x + 2)

yields

x^2 + 4x + 4

. Subtracting the entire second polynomial requires distributing the negative sign, resulting in

8x^2 + 10x - 3 - x^2 - 4x - 4

, which correctly simplifies to

7x^2 + 6x - 7

Step-by-Step Solution

Calculate the area of the rectangular panel.

(4x - 1)(2x + 3) = 8x^2 + 12x - 2x - 3 = 8x^2 + 10x - 3

The area of a rectangle is length multiplied by width, which requires multiplying two binomials using the FOIL method.

Calculate the area of the square opening.

(x + 2)^2 = (x + 2)(x + 2) = x^2 + 2x + 2x + 4 = x^2 + 4x + 4

The area of a square is its side length multiplied by itself.

Subtract the square's area from the rectangle's area.

(8x^2 + 10x - 3) - (x^2 + 4x + 4) = 8x^2 + 10x - 3 - x^2 - 4x - 4 = 7x^2 + 6x - 7

To find the remaining area, the removed portion must be subtracted. It is critical to distribute the negative sign to every term inside the parentheses being subtracted.

Key Concept

Multiplying binomials and subtracting polynomials while properly distributing negative signs.

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