Given, the complex numbers y² x - 5i and 4 + i(x + y² ) are conjugate to each other.

=> y² x - 5i and 4 - i(x + y² )

On comparing real and imaginary part, we get

y² x = 4   ............1

x + y² = 5  .........2

From equation 1, we get

x = 4/y²   ............3

Put value of x in equation 2, we get

      4/y² + y² = 5

=> (4 + y⁴ )/y² = 5

=> 4 + y⁴ = 5 * y²

=> y⁴ - 5y² + 4 = 0

=> y⁴ - 4y² - y² + 4 = 0

 => y² (y² - 4) - 1(y² - 4) = 0

=> (y² - 4) * (y² - 1) = 0

=> y² = 4, 1

=> y = ±2, ±1

Put value of y in equation 1, we get

      x = 4/(±2)² and x = 4/(±1)²

=> x = 4/4 and x = 4

=> x = 1, 4

Hence, when y = ±2, x = 1

and when when y = ±1, x = 4