$${C}=\mathrm{10}{m}^{\mathrm{3}} {p}^{\mathrm{2}} +{b}\left({m}^{\mathrm{3}} +\mathrm{6}{m}^{\mathrm{2}} {p}+\mathrm{3}{mp}^{\mathrm{2}} \right) \\$$$$\:+{c}\left(\mathrm{3}{m}^{\mathrm{2}} +\mathrm{6}{mp}+{p}^{\mathrm{2}} \right)+{d}\left(\mathrm{6}{m}+\mathrm{4}{p}\right) \\$$$$\:+\mathrm{10}{e} \\$$$${D}=\mathrm{10}{m}^{\mathrm{2}} {p}^{\mathrm{3}} +{b}\left(\mathrm{3}{m}^{\mathrm{2}} {p}+\mathrm{6}{mp}^{\mathrm{2}} +{p}^{\mathrm{3}} \right) \\$$$$\:+{c}\left({m}^{\mathrm{2}} +\mathrm{6}{mp}+\mathrm{3}{p}^{\mathrm{2}} \right)+{d}\left(\mathrm{4}{m}+\mathrm{6}{p}\right) \\$$$$\:+\mathrm{10}{e} \\$$$$\underset{−} {{C}−{D}=\mathrm{0}\:{gives}} \\$$$$\mathrm{10}{m}^{\mathrm{2}} {p}^{\mathrm{2}} \left({m}−{p}\right) \\$$$$+{b}\left[\left({m}^{\mathrm{3}} −{p}^{\mathrm{3}} \right)+\mathrm{6}{mp}\left({m}−{p}\right)−\mathrm{3}{mp}\left({m}−{p}\right)\right] \\$$$$+{c}\left[\mathrm{3}\left({m}^{\mathrm{2}} −{p}^{\mathrm{2}} \right)−\left({m}^{\mathrm{2}} −{p}^{\mathrm{2}} \right)\right] \\$$$$+{d}\left[\mathrm{6}\left({m}−{p}\right)−\mathrm{4}\left({m}−{p}\right)\right]+\mathrm{0}\:=\:\mathrm{0} \\$$$${If}\:{m}\:{would}\:{be}\:{p}\:{then}\:{z}={m}={p} \\$$$${otherwise}\:{dividing}\:{by}\:{m}−{p}\:{gives} \\$$$$\\$$$$\mathrm{10}{m}^{\mathrm{2}} {p}^{\mathrm{2}} +{b}\left({m}^{\mathrm{2}} +\mathrm{4}{mp}+{p}^{\mathrm{2}} \right) \\$$$$+{c}\left[\mathrm{2}\left({m}+{p}\right)\right]+\mathrm{2}{d}=\mathrm{0}\:\:\:\:…..\left({I}\right) \\$$$$\left({if}\:{m}\neq−{p}\:\:\:{even}\right) \\$$$${eq}.\left({i}\right)×\left({m}+{p}\right)\:{gives} \\$$$$\mathrm{10}{m}^{\mathrm{2}} {p}^{\mathrm{2}} \left({m}+{p}\right) \\$$$$+{b}\left[{m}^{\mathrm{3}} +\mathrm{4}{m}^{\mathrm{2}} {p}+{mp}^{\mathrm{2}} +{m}^{\mathrm{2}} {p}+\mathrm{4}{mp}^{\mathrm{2}} \right. \\$$$$\left.\:\:\:+{p}^{\mathrm{3}} \right]+{c}\left[\mathrm{2}{m}^{\mathrm{2}} +\mathrm{4}{mp}+\mathrm{2}{p}^{\mathrm{2}} \right] \\$$$$\:\:\:+{d}\left(\mathrm{2}{m}+\mathrm{2}{p}\right)=\mathrm{0}\:\:\:\:\:\:\:……\left({i}\right) \\$$$$\underset{−} {{Now}\:{C}+{D}=\mathrm{0}\:{gives}} \\$$$$\mathrm{10}{m}^{\mathrm{2}} {p}^{\mathrm{2}} \left({m}+{p}\right)+ \\$$$${b}\left[{m}^{\mathrm{3}} +{p}^{\mathrm{3}} +\mathrm{6}{mp}\left({m}+{p}\right)+\mathrm{3}{mp}\left({m}+{p}\right)\right] \\$$$$+{c}\left(\mathrm{4}{m}^{\mathrm{2}} +\mathrm{4}{p}^{\mathrm{2}} +\mathrm{12}{mp}\right) \\$$$$+{d}\left(\mathrm{10}{m}+\mathrm{10}{p}\right)+\mathrm{20}{e}\:=\mathrm{0}\:\:\:….\left({II}\right) \\$$$$\\$$$$\underset{−} {\left({II}\right)−\left({i}\right)\:{gives}} \\$$$$\:\:\boldsymbol{{b}}\left[\mathrm{4}\boldsymbol{{mp}}\left(\boldsymbol{{m}}+\boldsymbol{{p}}\right)\right]+ \\$$$$\:\boldsymbol{{c}}\left[\mathrm{2}\boldsymbol{{m}}^{\mathrm{2}} +\mathrm{8}\boldsymbol{{mp}}+\mathrm{2}\boldsymbol{{p}}^{\mathrm{2}} \right]+ \\$$$$\:\boldsymbol{{d}}\left[\mathrm{8}\left(\boldsymbol{{m}}+\boldsymbol{{p}}\right)\right]+\mathrm{20}\boldsymbol{{e}}=\mathrm{0} \\$$$$\Rightarrow \\$$$$\:\mathrm{2}{bmp}\left({m}+{p}\right)+{c}\left[\left({m}+{p}\right)^{\mathrm{2}} +\mathrm{2}{mp}\right] \\$$$$+\mathrm{4}{d}\left({m}+{p}\right)+\mathrm{10}{e}=\mathrm{0}\:\:\:\:….\left({III}\right) \\$$$$\\$$$${Now}\:{lets}\:{transform}\:{eq}.\:\left({I}\right)\:\&\left({III}\right) \\$$$${let}\:\:\boldsymbol{{m}}+\boldsymbol{{p}}=\boldsymbol{{s}}\:,\:{and}\:\boldsymbol{{mp}}=\boldsymbol{{g}} \\$$$$\underset{−} {\boldsymbol{{Eq}}.\:\left(\boldsymbol{{I}}\right)} \\$$$$\:\mathrm{10}\boldsymbol{{g}}^{\mathrm{2}} +\boldsymbol{{b}}\left(\boldsymbol{{s}}^{\mathrm{2}} +\mathrm{2}\boldsymbol{{g}}\right)+\mathrm{2}\boldsymbol{{cs}}+\mathrm{2}\boldsymbol{{d}}=\mathrm{0} \\$$$$\\$$$$\underset{−} {\boldsymbol{{Eq}}.\left(\boldsymbol{{III}}\right)} \\$$$$\:\:\mathrm{2}\boldsymbol{{bgs}}+\boldsymbol{{c}}\left(\boldsymbol{{s}}^{\mathrm{2}} +\mathrm{2}\boldsymbol{{g}}\right)+\mathrm{4}\boldsymbol{{ds}}+\mathrm{10}\boldsymbol{{e}}=\mathrm{0} \\$$$$\Rightarrow\:\:\boldsymbol{{g}}=−\frac{\boldsymbol{{cs}}^{\mathrm{2}} +\mathrm{4}\boldsymbol{{ds}}+\mathrm{10}\boldsymbol{{e}}}{\mathrm{2}\left(\boldsymbol{{bs}}+\boldsymbol{{c}}\right)} \\$$$$\underset{−} {{substituting}\:{for}\:\boldsymbol{{g}}\:{in}\:{eq}.\left({I}\right)} \\$$$$\mathrm{10}\left({cs}^{\mathrm{2}} +\mathrm{4}{ds}+\mathrm{10}{e}\right)^{\mathrm{2}} − \\$$$$\mathrm{4}{b}\left({cs}^{\mathrm{2}} +\mathrm{4}{ds}+\mathrm{10}{e}\right)\left({bs}+{c}\right) \\$$$$+\mathrm{4}\left({bs}^{\mathrm{2}} +\mathrm{2}{cs}+\mathrm{2}{d}\right)\left({bs}+{c}\right)^{\mathrm{2}} =\mathrm{0} \\$$$$\Rightarrow \\$$$$\mathrm{5}\left(\boldsymbol{{cs}}^{\mathrm{2}} +\mathrm{4}\boldsymbol{{ds}}+\mathrm{10}\boldsymbol{{e}}\right)^{\mathrm{2}} \\$$$$\:\:\:−\mathrm{2}\boldsymbol{{b}}\left(\boldsymbol{{bs}}+\boldsymbol{{c}}\right)\left(\boldsymbol{{cs}}^{\mathrm{2}} +\mathrm{4}\boldsymbol{{ds}}+\mathrm{10}\boldsymbol{{e}}\right) \\$$$$\:\:\:+\mathrm{2}\left(\boldsymbol{{bs}}+\boldsymbol{{c}}\right)^{\mathrm{2}} \left(\boldsymbol{{bs}}^{\mathrm{2}} +\mathrm{2}\boldsymbol{{cs}}+\mathrm{2}\boldsymbol{{d}}\right)=\mathrm{0} \\$$$${This}\:{is}\:{a}\:{degree}\:\mathrm{4}\:{eq}.\:{in}\:\boldsymbol{{s}}. \\$$$${So}\:{it}\:{seems}\:{i}\:{shall}\:{obtain}\:\boldsymbol{{s}} \\$$$${and}\:\boldsymbol{{g}}=−\frac{\boldsymbol{{cs}}^{\mathrm{2}} +\mathrm{4}\boldsymbol{{ds}}+\mathrm{10}\boldsymbol{{e}}}{\mathrm{2}\left(\boldsymbol{{bs}}+\boldsymbol{{c}}\right)}\:. \\$$$$\boldsymbol{{m}},\boldsymbol{{p}}\:=\:\frac{\boldsymbol{{s}}}{\mathrm{2}}\pm\sqrt{\frac{\boldsymbol{{s}}^{\mathrm{2}} }{\mathrm{4}}−\boldsymbol{{g}}} \\$$$${Now}\:{we}\:{then}\:{have} \\$$$$\:\:\:\boldsymbol{{At}}^{\mathrm{5}} +\boldsymbol{{Bt}}^{\mathrm{4}} +\boldsymbol{{Et}}+\boldsymbol{{F}}=\mathrm{0} \\$$$${or}\:\:{t}^{\mathrm{5}} +\left(\frac{{B}}{{A}}\right){t}^{\mathrm{4}} +\left(\frac{{E}}{{A}}\right){t}+\frac{{F}}{{A}}=\mathrm{0} \\$$$${but}\:{if}\:{m}={p}\:\:{clearly}\:{z}={m} \\$$$${hence}\:{all}\:{coefficients}\:{get}\:{zero} \\$$$${and}\:{we}\:{need}\:\:{to}\:{change}\: \\$$$${to}\:{z}=\frac{{mt}+{p}}{{t}−\mathrm{1}}\:;\:{this}\:{case}\:{i}\:{shall}\:{take} \\$$$${up}\:{in}\:{another}\:{post}.. \\$$$$\\$$
$$\mathrm{wow},\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}\:\mathrm{and}\:\mathrm{increase}\:\mathrm{your}\:\mathrm{knowledge}.\:\mathrm{I}\:\mathrm{will}\:\mathrm{learn} \\$$$$\mathrm{this}\:\mathrm{with}\:\mathrm{an}\:\mathrm{example}. \\$$