What is asked in the problem?
What is the integer?

Given:
Twice the square of a positive integer is 12 more than 10 times that integer.

Representation:
Let x = the integer

Write an Equation.Translate the sentence into mathematical equation

Twice the square of a positive integer is 12 more than 10 times that integer.
     2x^2 = 10x + 12

Solve the equation.
2x^2 - 10x - 12 = 0           Factor the equation to solve for x.
(2x + 2)(x - 6) = 0           Zero product Property.
2x + 2 = 2    or   x - 6 = 0
    2x = 0    or       x = 6
     x = 0    or       x = 6

Checking:
  2x^2 = 10x + 12, x= 0
2(0)^2 = 10(0) + 12
     0 = 0   ----------->> True

  2x^2 = 10x + 12, x=6
2(6)^2 = 10(6) + 12
     72 = 60 + 12
     72 = 72 --------->> True


Therefore the integer is 0 or 6.