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what describes the area of the figure as a simplified polynomial?

  9x² -15x +1

Step-by-step explanation:

The area can be figured as the difference of the enclosing rectangle area ((3x-2)×(4x-1)) and the area of the white space at the lower left (horizontal dimension x+1).

The vertical dimension of the white space at lower left is The difference of the given vertical dimensions:

  (4x -1) -(x -2) = 3x +1

So, the area of the figure is ...

  (3x -2)(4x -1) -(x +1)(3x +1)

  = (12x² -3x -8x +2) -(3x² +x +3x +1) . . . . "FOIL" each product

  = 12x² -3x -8x +2 -3x² -x -3x -1 . . . . . eliminate parentheses

  = x²(12 -3) +x(-3-8-1-3) +(2-1)

  = 9x² -15x +1

Comment on products of binomials

The product of any polynomial with any other can be found using the distributive property repeatedly. When both polynomials are binomials, a mnemonic can help you make sure all product terms are included.

  (a +b)(c +d) = a(c +d) + b(c +d) . . . . use the distributive property to eliminate the first set of parentheses

  = ac +ad +bc +bd . . . . . . . . . . . . . . use the distributive property to eliminate remaining parentheses

You will notice that the final product is the sum of the terms ...

  ac . . . product of First terms of the binomials

  ad . . . product of Outer terms of the expression

  bc . . . product of Inner terms of the expression

  bd . . . product of Last terms of the binomials

The letters of these descriptive words form the mnemonic FOIL, which is sometimes used as a verb (as above: FOIL each product).

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step-by-step explanation: