Simplify:
(i) 2x2(x3 − x) − 3x(x4 + 2x) − 2(x4 − 3x2)
(ii) x3y(x2 − 2x) + 2xy(x3 − x4)
(iii) 3a2 + 2(a + 2) − 3a(2a + 1)
(iv) x(x + 4) + 3x(2x2 − 1) + 4x2 + 4
(v) a(b − c) − b(c − a) − c(a − b)
(vi) a(b − c) + b(c − a) + c(a − b)
(vii) 4ab(a − b) − 6a2(b − b2) − 3b2(2a2 − a) + 2ab(b − a)
(viii) x2(x2 + 1) − x3(x + 1) − x(x3 − x)
(ix) 2a2 + 3a(1 − 2a3) + a(a + 1)
(x) a2(2a − 1) + 3a + a3 − 8
(xi) $\frac{3}{2} x^{2}\left(x^{2}-1\right)+\frac{1}{4} x^{2}\left(x^{2}+x\right)-\frac{3}{4} x\left(x^{3}-1\right)$
(xii) a2b(a − b2) + ab2(4ab − 2a2) − a3b(1 − 2b)
(xiii) a2b(a3 − a + 1) − ab(a4 − 2a2 + 2a) − b (a3 − a2 − 1)
(i) To simplify, we will use distributive law as follows:
$2 x^{2}\left(x^{3}-x\right)-3 x\left(x^{4}+2 x\right)-2\left(x^{4}-3 x^{2}\right)$
$=2 x^{5}-2 x^{3}-3 x^{5}-6 x^{2}-2 x^{4}+6 x^{2}$
$=2 x^{5}-3 x^{5}-2 x^{4}-2 x^{3}-6 x^{2}+6 x^{2}$
$=-x^{5}-2 x^{4}-2 x^{3}$
(ii) To simplify, we will use distributive law as follows:
$x^{3} y\left(x^{2}-2 x\right)+2 x y\left(x^{3}-x^{4}\right)$
$=x^{5} y-2 x^{4} y+2 x^{4} y-2 x^{5} y$
$=x^{5} y-2 x^{5} y-2 x^{4} y+2 x^{4} y$
$=-x^{5} y$
(iii) To simplify, we will use distributive law as follows:
$3 a^{2}+2(a+2)-3 a(2 a+1)$
$=3 a^{2}+2 a+4-6 a^{2}-3 a$
$=3 a^{2}-6 a^{2}+2 a-3 a+4$
$=-3 a^{2}-a+4$
(iv) To simplify, we will use distributive law as follows:
$x(x+4)+3 x\left(2 x^{2}-1\right)+4 x^{2}+4$
$=x^{2}+4 x+6 x^{3}-3 x+4 x^{2}+4$
$=x^{2}+4 x^{2}+4 x-3 x+6 x^{3}+4$
$=5 x^{2}+x+6 x^{3}+4$
(v) To simplify, we will use distributive law as follows:
$a(b-c)-b(c-a)-c(a-b)$
$=a b-a c-b c+b a-c a+c b$
$=a b+b a-a c-c a-b c+c b$
$=2 a b-2 a c$
(vi) To simplify, we will use distributive law as follows:
$a(b-c)+b(c-a)+c(a-b)$
$=a b-a c+b c-b a+c a-c b$
$=a b-b a-a c+c a+b c-c b$
$=0$
(vii) To simplify, we will use distributive law as follows:
$4 a b(a-b)-6 a^{2}\left(b-b^{2}\right)-3 b^{2}\left(2 a^{2}-a\right)+2 a b(b-a)$
$=4 a^{2} b-4 a b^{2}-6 a^{2} b+6 a^{2} b^{2}-6 b^{2} a^{2}+3 b^{2} a+2 a b^{2}-2 a^{2} b$
$=4 a^{2} b-6 a^{2} b-2 a^{2} b-4 a b^{2}+3 b^{2} a+2 a b^{2}+6 a^{2} b^{2}-6 b^{2} a^{2}$
$=-4 a^{2} b+a b^{2}$
(viii) To simplify, we will use distributive law as follows:
$x^{2}\left(x^{2}+1\right)-x^{3}(x+1)-x\left(x^{3}-x\right)$
$=x^{4}+x^{2}-x^{4}-x^{3}-x^{4}+x^{2}$
$=x^{4}-x^{4}-x^{4}-x^{3}+x^{2}+x^{2}$
$=-x^{4}-x^{3}+2 x^{2}$
(ix) To simplify, we will use distributive law as follows:
$2 a^{2}+3 a\left(1-2 a^{3}\right)+a(a+1)$
$=2 a^{2}+3 a-6 a^{4}+a^{2}+a$
$=2 a^{2}+a^{2}+3 a+a-6 a^{4}$
$=3 a^{2}+4 a-6 a^{4}$
(x) To simplify, we will use distributive law as follows:
$a^{2}(2 a-1)+3 a+a^{3}-8$
$=2 a^{3}-a^{2}+3 a+a^{3}-8$
$=2 a^{3}+a^{3}-a^{2}+3 a-8$
$=3 a^{3}-a^{2}+3 a-8$
(xi) To simplify, we will use distributive law as follows:
$\frac{3}{2} x^{2}\left(x^{2}-1\right)+\frac{1}{4} x^{2}\left(x^{2}+x\right)-\frac{3}{4} x\left(x^{3}-1\right)$
$=\frac{3}{2} x^{4}-\frac{3}{2} x^{2}+\frac{1}{4} x^{4}+\frac{1}{4} x^{3}-\frac{3}{4} x^{4}+\frac{3}{4} x$
$=\frac{3}{2} x^{4}+\frac{1}{4} x^{4}-\frac{3}{4} x^{4}+\frac{1}{4} x^{3}-\frac{3}{2} x^{2}+\frac{3}{4} x$
$=\left(\frac{6+1-3}{4}\right) x^{4}+\frac{1}{4} x^{3}-\frac{3}{2} x^{2}+\frac{3}{4} x$
$=x^{4}+\frac{1}{4} x^{3}-\frac{3}{2} x^{2}+\frac{3}{4} x$
(xii) To simplify, we will use distributive law as follows:
$a^{2} b\left(a-b^{2}\right)+a b^{2}\left(4 a b-2 a^{2}\right)-a^{3} b(1-2 b)$
$=a^{3} b-a^{2} b^{3}+4 a^{2} b^{3}-2 a^{3} b^{2}-a^{3} b+2 a^{3} b^{2}$
$=a^{3} b-a^{3} b-a^{2} b^{3}+4 a^{2} b^{3}-2 a^{3} b^{2}+2 a^{3} b^{2}$
$=3 a^{2} b^{3}$
(xiii) To simplify, we will use distributive law as follows:
$a^{2} b\left(a^{3}-a+1\right)-a b\left(a^{4}-2 a^{2}+2 a\right)-b\left(a^{3}-a^{2}-1\right)$
$=a^{5} b-a^{3} b+a^{2} b-a^{5} b+2 a^{3} b-2 a^{2} b-a^{3} b+a^{2} b+b$
$=a^{5} b-a^{5} b-a^{3} b+2 a^{3} b-a^{3} b+a^{2} b-2 a^{2} b+a^{2} b+b$
$=b$