Is there some classical technique (from algebra or analysis) to find the expressions of roots of quintic and sextic using the root expressions of quadratic, cubic and quartic root expressions.
For example: An instance in the case of general quintic would be \begin{align*}x^5+bx^4+cx^3+dx^2+ex+f&= (x-q_1)(x-q_2)(x-q_3)(x-q_4)(x-q_5)\\ &=(x^2+ \beta_1x+\gamma_1)(x^3+ \beta_2x^2+\gamma_2x+\gamma_3)\\ &=(x-r_1)(x-r_2)(x-c_1)(x-c_2)(x-c_3). \end{align*}
The goal is to write $(\forall~i)~q_i$ using roots $r_i$ of quadratic (given in coefficients $\beta_1,\gamma_1$ as $r_{1,2}=(-\beta_1\pm \sqrt{\beta_1^2-4\gamma_1})/2)$, and the roots of the cubic $c_i$ (given in coefficients $\beta_2,\gamma_2,\gamma_3$). And a (maybe never) possible real root could look like $$q_1=^{?} \Big(\frac{-\beta_1+ \sqrt{\beta_1^2-4\gamma_1}}{2}\Big)\times^{?} \bigg(-\frac{-\beta_2}{3}+\sqrt[3]{\frac{-q}{2}+\sqrt{\Delta}}-\sqrt[3]{\frac{-q}{2}+\sqrt{\Delta}}\bigg),$$
where $q$ and $\Delta$ are given in coefficients $\beta_2,\gamma_2,\gamma_3$(skipping for brevity.)
Are there any particular cases where this decomposition is possible, keeping in mind no radical expressions of roots exists for quintics? Please let me know: any hints and references are appreciated.
Also, welcome to the suggestions to improve the clarity of this silly question: thank you in advance.
Disclaimer: It might sound absurd for now mathematically because there is no formula for roots of general quintic. But putting it out if ever possible?