4. Solve algebraically \( 4 x^{3}-20 x^{2}>-17 x-14 \)

To solve the inequality \( 4x^{3} - 20x^{2} > -17x - 14 \) algebraically, first, we can rearrange it to one side: \[ 4x^{3} - 20x^{2} + 17x + 14 > 0. \] Next, combine like terms to form a polynomial inequality: \[ 4x^{3} - 20x^{2} + 17x + 14 = 0. \] Finding the roots of the polynomial can be complex, so we can look for rational roots using the Rational Root Theorem. Testing possible values, we find \( x = 2 \) is a root. We can use synthetic division or polynomial long division to factor the cubic polynomial down, typically yielding a quadratic. Once factored, we analyze the sign of the polynomial by testing intervals on the number line defined by the roots and critical points. The eventual solution will be the intervals where the polynomial is greater than zero. Remember to verify your intervals by choosing test points!