Answer:
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
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<em>Comment on products of binomials</em>
The product of any polynomial with any other can be found using the distributive property repeatedly. When both polynomials are <em>binomials</em>, 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).
