$a)$

$S = x^{3} - x + 2 x^{4} - 2 = 2 x^{4} + x^{3} - x - 1$

$T = 4 - x^{3} + 3 x^{2} + x^{4} = x^{4} - x^{3} + 3 x^{2} + 4$

$S + T = (2 x^{4} + x^{3} - x - 1) + (x^{4} - x^{3} + 3 x^{2} + 4) =$

$= \underline{2 x^{4}} + \underline{\underline{x^{3}}} - x - \underline{\underline{\underline{1}}} + \underline{x^{4}} - \underline{\underline{x^{3}}} + 3 x^{2} + \underline{\underline{\underline{4}}} =$

$= 3 x^{4} + 3 x^{2} - x + 3$

$S - T = (2 x^{4} + x^{3} - x - 1) - (x^{4} - x^{3} + 3 x^{2} + 4) =$

$= \underline{2 x^{4}} + \underline{\underline{x^{3}}} - x - \underline{\underline{\underline{1}}} - \underline{x^{4}} + \underline{\underline{x^{3}}} - 3 x^{2} - \underline{\underline{\underline{4}}} =$

$= x^{4} + 2 x^{3} - 3 x^{2} - x - 5$

$b)$

$S = - 2 x^{2} + 4 x - x^{6} + 2 = - x^{6} - 2 x^{2} + 4 x + 2$

$T = - 5 x + 3 x^{2} + x^{6} - 3 x^{5} = x^{6} - 3 x^{5} + 3 x^{2} - 5 x$

$S + T = (- x^{6} - 2 x^{2} + 4 x + 2) + (x^{6} - 3 x^{5} + 3 x^{2} - 5 x) =$

$= \underline{- x^{6}} - \underline{\underline{2 x^{2}}} + \underline{\underline{\underline{4 x}}} + 2 + \underline{x^{6}} - 3 x^{5} + \underline{\underline{3 x^{2}}} - \underline{\underline{\underline{5 x}}} =$

$= - 3 x^{5} + x^{2} - x + 2$

$S - T = (- x^{6} - 2 x^{2} + 4 x + 2) - (x^{6} - 3 x^{5} + 3 x^{2} - 5 x) =$

$= \underline{- x^{6}} - \underline{\underline{2 x^{2}}} + \underline{\underline{\underline{4 x}}} + 2 - \underline{x^{6}} + 3 x^{5} - \underline{\underline{3 x^{2}}} + \underline{\underline{\underline{5 x}}} =$

$= - 2 x^{6} + 3 x^{5} - 5 x^{2} + 9 x + 2$

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