Acarpenter is assigned a job of installing a spa into a pre-existing deck. the dimensions of the deck are (5x + 2) wide by (3x + 1) long. the dimensions of the spa are (2x + 3) wide by (x + 2) long.

write the polynomial that represents the remaining area of the deck after the carpenter cut the hole out for the spa.

find the area of the new deck if x = 2 feet.

The dimensions of the rectangular deck are (5x + 2) wide by (3x + 1) long.
The area of the deck is
A₁ = (5x + 2)(3x + 1)
     = 15x² + 5x + 6x + 2
     = 15x² + 11x + 2
15x² + 11x + 2

The dimensions of the spa are (2x + 3) wide by (x + 2) long.
The area of the spa is
A₂ = (2x + 3)(x + 2)
     = 2x² + 4x + 3x + 6
     = 2x² + 7x + 6
2x² + 7x + 6

The area remaining after the hole for the spa has been cut from the deck(the area of the new deck) is
A = A₁ - A₂
   = 15x² + 11x + 2 - (2x² + 7x + 6)
   = 15x² - 2x² + 11x - 7x + 2 - 6
   = 13x² + 9x - 4
13x² + 9x - 4

If x = 2 feet, the area of the new deck is
A(2) = 13*2² + 9*2 - 4  = 78 ft²
78 ft²

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