Question

Simplify:( 2x - 2)( 3x^2 + 9x - 3). What is the coefficient of the quadratic term?

Answer 1

You can use the distributive property and multiply this out.
  2x(3x² +9x -3) -2(3x² +9x -3)
  = 6x³ +18x²-6x -6x² -18x +6
  = 6x³ +12x² -24x +6

The coefficient of the x² term is 12.

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You can do this in your head using the techniques of Vedic mathematics. Write the coefficients of the two expressions, putting 0 for the x² coefficient in the first factor.
  0 2 -2
  3 9 -3

The x² term of the product will be the sum of the products of terms whose degrees add to 2. That is, the first term of the first row times the last term of the second row, plus the product of the middle two numbers in each row, plus the product of the first term in the second row times the last term in the first row.

Here's a more visual representation of the products that get added to get the coefficient of the x² term.
  0 2 -2
  3 9 -3 . . . . product is 0

  0 2 -2
  3 9 -3 . . . . product is 18

  0 2 -2
  3 9 -3 . . . . product is -6

The sum, of course, is 0 +18 -6 = 12