If $\mathrm{l}, \mathrm{m}, \mathrm{n}$ are real and $\mathrm{l}\ne \mathrm{m},$ then the roots of the equation $(\mathrm{l}-\mathrm{m}){\mathrm{x}}^2-5(\mathrm{l}+\mathrm{m})\mathrm{x}-2(\mathrm{l}-\mathrm{m})=0$ are 

Given equation is $(\mathrm{l}-\mathrm{m}){\mathrm{x}}^2-5(\mathrm{l}+\mathrm{m})\mathrm{x}-2(\mathrm{l}-\mathrm{m})=0$

Its discriminant $\mathrm{D}={(\mathrm{l}+\mathrm{m})}^2+8{(\mathrm{l}-\mathrm{m})}^2$

Which is positive, since $\mathrm{l}, \mathrm{m}, \mathrm{n}$ are real and $\mathrm{l}\ne \mathrm{m}.$

Hence roots are real and distinct.