To solve this equation using D5 = ±1; ±5 method, we need to substitute x² with y, thus making the equation into a quadratic form.

Let y = x²

Therefore, the equation becomes:

12y² + 19y + 5 = 0

We can factorize this quadratic equation:

(4y + 1)(3y + 5) = 0

Now, we solve for y:

4y + 1 = 0 or 3y + 5 = 0

4y = -1 3y = -5

y = -1/4 y = -5/3

Now, since y = x², we substitute back in:

For y = -1/4: x² = -1/4

x = √(-1/4) or x = -√(-1/4)

x = ±1/2i or x = ±1/2i

For y = -5/3: x² = -5/3

x = √(-5/3) or x = -√(-5/3)

x = ±√(5/3)i or x = ±√(5/3)i

Therefore, the solutions to the equation 12x⁴ + 19x² + 5 = 0 are:

x = ±1/2i, ±√(5/3)i.