37th Hellenic Mathematical Olympiad 2020

Question

Solve in the real numbers the inequality: (x+2)^4x^3 - (x+2)^22x -x16.

Accepted Answer

For x 0 we have: align* (x+2)^4x^3 - (x+2)^22x & -x16 (x+2)^4x^3 - (x+2)^22x + x16 0 & 16(x+2)^4 - 8(x+2)^2 x^2 + x^416x^3 0 (4(x+2)^2 - x^2)^216x^3 0 & x(4(x+2)^2 - x^2)^2 0, x 0 x 0, x 0 or 4(x+2)^2 - x^2 = 0 & x > 0 or (2x+4+x)(2x+4-x) = 0 x > 0 or (3x+4)(x+4) = 0 & x > 0 or x = -43 or x = -4. align*