Final answer:
The real solutions of the polynomial equation 4x³ + 2x² - 15x - 6 = 0 are -3/2, -1/4, and 2.
Step-by-step explanation:
To find the real solutions of the polynomial equation 4x³ + 2x² - 15x - 6 = 0, we can use the Rational Root Theorem to determine possible rational roots.
By testing the factors of the constant term (-6) divided by the factors of the leading coefficient (4), we find that x = -3/2 is a root.
Using synthetic division, we divide the polynomial by (2x + 3) to get a quotient of 4x² - 5x - 2.
This quadratic equation can then be factored or solved using the quadratic formula to find the remaining roots.
Factorizing the quadratic equation 4x² - 5x - 2 = 0, we get (4x + 1)(x - 2) = 0.
So, the roots are x = -3/2, x = -1/4, and x = 2.
Therefore, the real solutions of the given polynomial equation are x = -3/2, x = -1/4, and x = 2.