x^3-12 x+16-2(x+1) sqrt(5-4 x-x^2)=0

Let the equation be<br />$$x^3 - 4.2x + 16 - 2(2x+1)\sqrt{7+4x} = 0$$<br />$$x^3 - 4.2x + 16 = 2(2x+1)\sqrt{7+4x}$$<br />Square both sides:<br />$$(x^3 - 4.2x + 16)^2 = 4(2x+1)^2(7+4x)$$<br />This leads to a sixth-degree polynomial equation. Solving this analytically is extremely difficult, if not impossible. Numerical methods (like Newton-Raphson) are required to find approximate solutions. Software or a calculator capable of numerical equation solving should be used.<br />