see, that polynomial is the polynomial of $7^{th}$ degree $$\label{Ineq_21}
h_3(x)
=
P_7(x)
=
4 x-4\left(\nfrac{8}{45}-a(x) \right) x^5
=
\nfrac{32}{945} x^7 - \nfrac{32}{45} x^5 + 4 x.$$ Using the factorization of the polynomial $P_7(x)$ we have $$\label{Ineq_22}
P_7(x)=\nfrac{4}{945} x(8 x^6 -168 x^4+945)=\nfrac{4}{945} x P_6(x) ,$$ where $$\label{Ineq_23}
P_6(x)=8 x^6 -168 x^4+945,$$ for $x\!\in\!(0,1.57]$. Introducing the substitution $s= x^2$ we can notice that the polynomial $P_6(x)$ can be transformed into polynomial of $3^{rd}$ degree $$\label{Ineq_24}
P_3(s)=8 s^3-168 s^2+945,$$ for $s\!\in\!(0,2.4649]$. Using MATLAB software we can determine the real numerical factorization of the polynomial $$\label{Ineq_25}
P_3(s)=\alpha(s - s_1)(s - s_2)(s - s_3),$$ where $\alpha = 8$ and where

$$\label{Ineq_26}
\begin{array}{rcl}
s_1 \!\!&\!=\!&\!\!\!-\nfrac{1}{8}\left(18172\!+\!84 I \sqrt{21495}\right)^{1/3}\!-\!\nfrac{98}{\left(18172\!+\!84 I \sqrt{21495}\right)^{1/3}} \!+\! 7                                \\[-0.1 ex]
\!\!&\! \!&\!\!\!+\nfrac{3}{4} I \sqrt{3}\left(\nfrac{1}{6}\left(18172\!+\!84 I \sqrt{21495}\right)^{1/3} \!-\! \nfrac{392}{3\left(18172 \!+\! 84 I \sqrt{21495}\right)^{1/3}}\right)  \\[0.00 ex]
\!\!&\!=\!&\!\!-2.253\ldots ,                                                                                                                                                          \\[1.50 ex]
s_2 \!\!&\!=\!&\!\! \!-\nfrac{1}{8}\left(18172\!+\!84 I \sqrt{21495}\right)^{1/3}\!-\!\nfrac{98}{\left(18172\!+\!84 I \sqrt{21495}\right)^{1/3}} \!+\! 7                               \\[-0.1 ex]
\!\!&\! \!&\!\!\!-\nfrac{3}{4} I \sqrt{3}\left(\nfrac{1}{6}\left(18172\!+\!84 I \sqrt{21495}\right)^{1/3} \!-\! \nfrac{392}{3\left(18172 \!+\! 84 I \sqrt{21495}\right)^{1/3}}\right)  \\[0.00 ex]
\!\!&\!=\!&\!\! 2.528\ldots ,                                                                                                                                                          \\[1.50 ex]
s_3 \!\!&\!=\!&\!\!\nfrac{1}{4}\left(18172\!+\!84 I \sqrt{21495}\right)^{1/3} \!+\! \nfrac{196}{\left(18172 \!+\! 84 I \sqrt{21495}\right)^{1/3}} \!+\!7                               \\[0.00 ex]
\!\!&\!=\!&\!\! 20.724\ldots ;                                                                                                                                                         \\[1.50 ex]
\end{array}$$ for $I= \sqrt{-1}$ (imaginary unit). The polynomial $P_3(s)$ has exactly three simple real roots with 