a) P(x) = (4x + 1 - $x^{2}$ + 2$x^{3}$) - ( $x^{4}$ + 3x - $x^{3}$ - 2$x^{2}$ -5)
= 4x + 1 - $x^{2}$ + 2$x^{3}$ - $x^{4}$ - 3x + $x^{3}$ + 2$x^{2}$ +5
= -$x^{4}$ + $3x^{3}$ + $x^{2}$ + x + 6
Q(x) = $3x^{4}$ + $2x^{5}$ - 3x - $5x^{4}$ - $x^{5}$ + x + $2x^{2}$ - 1
= $x^{5}$ - 2$x^{4}$ + $2x^{2}$ - 2x + 1
b) P(x) + Q(x) = -$x^{4}$ + $3x^{3}$ + $x^{2}$ + x + 6 + $x^{5}$ - 2$x^{4}$ + $2x^{2}$ - 2x + 1
= $x^{5}$ - $x^{4}$ + $3x^{3}$ + $3x^{2}$ - x + 7
P(x)-Q(x) = -$x^{4}$ + $3x^{3}$ + $x^{2}$ + x + 6 - ($x^{5}$ - 2$x^{4}$ + $2x^{2}$ - 2x + 1)
= -$x^{4}$ + $3x^{3}$ + $x^{2}$ + x + 6 - $x^{5}$ + 2$x^{4}$ - $2x^{2}$ + 2x - 1
= -$x^{5}$ + $x^{4}$ + $3x^{3}$ - $x^{2}$ + 3x + 5