$a)-7x^3{y}^2+3x-9x^{2}y+2x^3{y}^2-6x+13x^{2}y+x=-7x^3{y}^2+2x^3{y}^2+3x-6x+x-9x^{2}y+13x^{2}y=-5x^3{y}^2-2x+4x^{2}y=-5x^3{y}^2+4x^{2}y-2x$  

$b)12x^{2}y-3xy+6x^2{y}^2-12x{y}^2+4xy-8x^2{y}^2+6x{y}^2-19x^{2}y+8x^2{y}^2=12x^{2}y-19x^{2}y-3xy+4xy+6x^2{y}^2-8x^2{y}^2+8x^2{y}^2-12x{y}^2+6x{y}^2=-7x^{2}y+xy+6x^2{y}^2-6x{y}^2$  

$c)-5x^5+x+7x^4-2-3x+18x^5+21x^2-3x^4+7x=-5x^5+18x^5+7x^4-3x^4+21x^2+x-3x+7x-2=13x^5+4x^4+21x^2+5x-2$  

$d){\left(-2x^2+y\right)}^2-{\left(x+2y^2\right)}^2-\left(y-2x^2\right)\left(x+2y^2\right)+2\left(y^3-x^3\right)-4\left(x^4-y^4\right)=4x^4-4x^{2}y+y^2-\left(x^2+4x{y}^2+4y^4\right)-\left(yx+2y^3-2x^3-4x^2{y}^2\right)+2y^3-2x^3-4x^4+4y^4=4x^4-4x^{2}y+y^2-x^2-4x{y}^2-4y^4-yx-2y^3+2x^3+4x^2{y}^2+2y^3-2x^3-4x^4+4y^4=$  

Wyrazy podobne:

$4x^4-4x^4=0$  

$-4x^{2}y$  

$y^2$  

$-x^2$  

$-4x{y}^2$  

$-4y^4+4y^4=0$  

$-xy$  

$-2y^3+2y^3=0$  

$2x^3-2x^3=0$  

$4x^2{y}^2$  

A więc:

$-4x^{2}y+y^2-x^2-4x{y}^2-xy+4x^2{y}^2$  

e) Wykonajmy potęgowanie poszczególnych wyrażeń:

${\left(x^2-1\right)}^3=x^6-3x^4+3x^2-1$   

$3x^2\left(2x-x^2\right)=6x^3-3x^4$  

${\left(2x+1\right)}^3=8x^3+3\cdot 4x^2+3\cdot 2x\cdot 1+1=8x^3+12x^2+6x+1$  

Zatem:

$x^6-3x^4+3x^2-1-\left(6x^3-3x^4\right)+8x^3+12x^2+6x+1=x^6-3x^4+3x^2+1-6x^3+3x^4+8x^3+12x^2+6x+1$  

Wyrazy podobne:

$x^6$  

$-3x^4+3x^4=0$  

$-6x^3+8x^3=2x^3$  

$3x^2+12x^2=15x^2$  

$6x$  

$-1+1=0$  

Rozwiązanie to:

$x^6+2x^3+15x^2+6x$