Question

Alexei wants to hang a mirror in his boat but the mirror and frame must not have an area larger than 19.25 square feet. The mirror is 3 feet wide and 5 feet long. Which quadratic equation represents the area of the mirror and frame combined? (The border must have equal width on each side.)
Answer
2x^2 + 8x - 19.25 = 0
4x^2 + 16x + 15 = 0
4x^2 + 16x - 4.25 = 0
3x^2 + 5x - 19.25 = 0

Answer 1

Let x = width of the border in feet

A = (3+2x)(5+2x) = 15 + 16x + 4x² ≤ 19.25

4(x² + 4x + 4) - 16 + 15 ≤ 19.25
4(x+2)² ≤ 20.25
(x+2)² ≤ 5.0625 = 81/16

-9/4 ≤ x+2 ≤ 9/4
-17/4 ≤ x ≤ 1/4

Since a negative width border is outside the domain, discard the interval where x < 0

The border width must be in the interval of 0 ≤ x ≤ 1/4 foot (3 inches) for the total area of the mirror and frame to be not larger than 19.25 ft²

Hope this helps!
if confusing i will explain another if you want too.

Answer 2

here is another examples:

The third one :
the width of mirror plus 2 frames is (3 + 2x)
the length is 5 +2x
Area = (Len )(width ) = 19.25
= (3+2x)(5+2x) = 15 + 10x + 6x + 4x^2 = 19.25
or
4x^2 +16x -4.25 <---- answer " C"