Let $0\le x\le 1$, show that inequality $$99x^7-381x^6+225x^5-415x^4+157x^3-3x^2-x-1\le 0$$
This problem comes from the fact that I solved a different inequality.I tried to solve it by factorizing it to see if I could get symbols.but I failed.
this inequality is hold by wolfapha Test it.
Because by AM-GM $$1+x+3x^2-157x^3+415x^4-225x^5+381x^6-99x^7=$$ $$=1+x+3x^2-157x^3+370x^4+x^4(45-225x+282x^2)+99x^6(1-x)\geq$$ $$\geq370x^4+3x^2+x+1-157x^3=6\cdot\frac{185}{3}x^4+3x^2+x+1-157x^3\ge$$ $$\geq9\sqrt[9]{\left(\frac{185}{3}x^4\right)^6\cdot3x^2\cdot x\cdot1}-157x^3=\left(9\sqrt[9]{\left(\frac{185}{3}\right)^6\cdot3}-157\right)x^3\geq0.$$