2 distinct real roots

\hspace{-16}$Here $\mathbf{(x^2-5x+4)^2+(2m+1).(x^2-5x+4)+3=0}$\\\\ Now Let $\mathbf{{x^2-5x+4}=t}\;$, Then equation Convert into \\\\ $\mathbf{t^2+(2m+1).t+3=0}$\\\\ So here $\mathbf{D=(2m+1)^2-4.3=4m^2+4m-11}$\\\\ $\boxed{\mathbf{D=4m^2+4m-11}}$\\\\ So $\mathbf{t=\frac{-(2m+1)\pm \sqrt{D}}{2}}$\\\\ $\mathbf{x^2-5x+4=\frac{-(2m+1)\pm \sqrt{D}}{2}}$\\\\ Now Taking $\mathbf{(+ve)}$ sign and $\mathbf{(-ve)}$ sign seperately \\\\ $\mathbf{x^2-5x+4=\frac{-(2m+1)+\sqrt{D}}{2}}$ and $\mathbf{x^2-5x+4=\frac{-(2m+1)-\sqrt{D}}{2}}$\\\\ \textbf{\underline{\underline{For First::::}}}\\\\ $\mathbf{2x^2-10x+8=-(2m+1)+\sqrt{D}\Leftrightarrow 2x^2-10x+\left(9+2m-\sqrt{D}\right)=0}$\\\\ So For Distinct real Roots either Discriminant is $\mathbf{> 0}$ OR $\mathbf{<0}$\\\\ Let $\mathbf{D_{1}=100-4.2.\left(9+2m-\sqrt{D}\right)=100-72-16m+8\sqrt{D}}$\\\\ $\boxed{\mathbf{D_{1}=28-16m+8\sqrt{D}=4.(7-4m+2\sqrt{D})}}$\\\\ Now If $\mathbf{D_{1}>0\Leftrightarrow 7-4m+2\sqrt{D}>0}$\\\\

\hspace{-16}$So $\mathbf{2\sqrt{D}>-\left(7-4m\right)}$\\\\ $\mathbf{2\sqrt{4m^2+4m-11}>-(7-4m)}$\\\\ $\bullet$ If $\mathbf{(7-4m)\geq 0\Leftrightarrow m\geq \frac{7}{4}}$\\\\ So Given Inequality is Valid for $\mathbf{\boxed{\bold{m\geq \frac{7}{4}}}...............(1)}$ \\\\ $\bullet$ If $\mathbf{(7-4m)< 0\Leftrightarrow m< \frac{7}{4}}$\\\\\\ So $\mathbf{2\sqrt{4m^2+4m-11}>-(7-4m)\Leftrightarrow 4.(4m^2+4m-11)<(49+16m^2-56m)}$\\\\ So $\mathbf{m> \frac{93}{72}}$\\\\ So $\mathbf{\boxed{\frac{93}{72}<\bold{m< \frac{7}{4}}}..............(2)}$\\\\ So from equation $\mathbf{(1)}$ and $\mathbf{(2)}$ $\mathbf{\boxed{\bold{m> \frac{93}{74}}}}$ for $\mathbf{D_{1}>0}$\\\\ Now You can calculate for $\mathbf{D_{1}<0}$

\hspace{-16}$\textbf{\underline{\underline{For Second::::}}}\\\\ $\mathbf{2x^2-10x+8=-(2m+1)-\sqrt{D}\Leftrightarrow 2x^2-10x+\left(9+2m+\sqrt{D}\right)=0}$\\\\ So For Distinct real Roots either Discriminant is $\mathbf{> 0}$ OR $\mathbf{<0}$\\\\ Let $\mathbf{D_{2}=100-4.2.\left(9+2m+\sqrt{D}\right)=100-72-16m-8\sqrt{D}}$\\\\ $\boxed{\mathbf{D_{2}=28-16m-8\sqrt{D}=4.(7-4m-2\sqrt{D})}}$\\\\ Now If $\mathbf{D_{2}>0\Leftrightarrow 7-4m-2\sqrt{D}>0}$\\\\ $\mathbf{(7-4m)>2\sqrt{D}\Leftrightarrow 2\sqrt{D}<(7-4m)}$\\\\ $\bullet$ If $\mathbf{(7-4m)\leq 0 }$ and $\mathbf{D=0}$\\\\ No value of $\mathbf{m}$\\\\ $\bullet$ If $\mathbf{(7-4m)> 0\Leftrightarrow \mathbf{m<\frac{7}{4}} }$\\\\ $\mathbf{2\sqrt{D}<(7-4m)}$\\\\ So $\mathbf{m< \frac{93}{72}}$\\\\ So Second Inequality is valid for $\mathbf{\boxed{\bold{m <\frac{93}{74}}}}$\\\\ So $\mathbf{\boxed{\bold{m <\frac{93}{74}}}}$ for $\mathbf{D_{2}>0}$\\\\\\ Now You can calculate for $\mathbf{D_{2}<0}$\\\\\\

\hspace{-16}$Now Given equation has $\mathbf{2}$ distinct Real Roots , Then \\\\ $\bullet$ $\mathbf{D_{1}>0}$ and $\mathbf{D_{2}<0}$\\\\ $\bullet$ $\mathbf{D_{1}<0}$ and $\mathbf{D_{2}>0}$\\\\

agar galti ho to dekh lena yaar....

thanks

Thanks! Itni mehnat ke baad galti ki jegah kaha!!

But is there any other method available for doing this sum? HS sir Kindly let us know!