The square of negative nonzero integer is equal to four times of its absolute value. What is the integer?

To find the integer that satisfies the given conditions, we can set up an equation.

Let's assume that the integer is represented by 'x'. According to the given information, the square of the negative nonzero integer is equal to four times its absolute value:

(x)^2 = 4 * |x|

To solve this equation, we need to consider two cases: when x is positive and when x is negative.

1. When x is positive:
If x is positive, then |x| is equal to x. Therefore, the equation can be rewritten as:
x^2 = 4x

Rearranging the equation:
x^2 - 4x = 0

Factoring out an 'x':
x(x - 4) = 0

Setting each factor equal to zero:
x = 0 or x - 4 = 0

From this, we find two possible solutions: x = 0 or x = 4.

2. When x is negative:
If x is negative, then |x| is equal to -x. Therefore, the equation can be rewritten as:
x^2 = 4(-x)

Expanding and rearranging the equation:
x^2 = -4x

Adding 4x to both sides:
x^2 + 4x = 0

Factoring out an 'x':
x(x + 4) = 0

Setting each factor equal to zero:
x = 0 or x + 4 = 0

From this, we find two possible solutions: x = 0 or x = -4.

So, the possible solutions to the equation are x = 0, x = 4, and x = -4. However, we can see that the given condition specifies a negative nonzero integer. Therefore, the only solution that satisfies this condition is x = -4.