general method of solving simultaneous equ. and elimination of variable

given $\displaystyle x^4+ax^3+bx^2-8x+4$ is the identity of

$\displaystyle x^4 +(l+c)x^3+(m+d+cl)x^2 +(cm+dl)x + dm $

then $\displaystyle l+c=a $

$\displaystyle m+d+cl=b$

$\displaystyle cm+dl=-8$

$\displaystyle dm=4$

i have to find abcdlm. i always write so many useless steps in solving simultaneous equ by elimination. would any one give me some advice and a general guide so that i could do it quick and fast without writing redundant steps? thanks.