<s>[INST] solving the equation $x^4-5x^3+11x^2-13x+6=0$ by given condition 1.(a) solve the equation $x^4-5x^3+11x^2-13x+6=0$ , given that two of its roots $p$ & $q$ are connected by the relation $3p+2q=7$
(b) solve the equation $x^4-5x^3+11x^2-13x+6=0$ which has two roots whose difference is $1$
did I need to solve these problems by  taking roots $x_1,x_2,x_3,x_4$ and using the relation between roots and coefficients and the given facts. it will be then a very lengthy process.is there any alternative short process
 [/INST] $$x^4-5x^3+11x^2-13x+6=0$$
$$x^4-x^3-4x^3+4x^2+7x^2-7x-6x+6=0$$
$$(x-1)x^3-(x-1)4x^2+(x-1)7x-(x-1)6=0$$
$$(x-1)(x^3-4x^2+7x-6)=0$$
$$(x-1)(x^3-2x^2-2x^2+4x+3x-6)=0$$
$$(x-1)\left( (x-2)x^2-(x-2)2x+(x-2)3 \right)=0$$
$$(x-1)(x-2)(x^2+2x+3)=0$$
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