Brice wants to build a table and put a border around it. the table and border must have an area of 3,006 square inches. the table is 36 inches wide and 72 inches long without the border. which quadratic equation can be used to determine the thickness of the border, x? a. 4x2 + 216x + 2,592 = 0 b. 4x2 + 216x − 414 = 0 c. 2x2 + 216x − 3,004 = 0 d. x2 + 108x + 2,592 = 0

B is the correct answer.

We are adding the border of x thickness to all 4 sides of the table.  This makes the new width of the table 36+x+x = 36+2x and the new length of the table 72+x+x=72+2x.  To find the area, we multiply the length and width:

(36+2x)(72+2x) = 3006

Multiplying the binomials, we get:
36*72 + 36*2x + 2x*72 + 2x*2x = 3006
2592 + 72x + 144x + 4x² = 3006

Combining like terms gets us
2592 + 216x + 4x² = 3006

Subtract 3006 from both sides:
2592 + 216x + 4x² - 3006 = 3006 - 3006
-414 + 216x + 4x² = 0

Writing in standard form we have
4x²+216x-414=0