Simplify:

Solution:

(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:​

$a^{2} b\left(a^{3}-a+1\right)-a b\left(a^{4}-2 a^{2}+2 a\right)-b\left(a^{3}-a^{2}-1\right)$

$=a^{5} b-a^{3} b+a^{2} b-a^{5} b+2 a^{3} b-2 a^{2} b-a^{3} b+a^{2} b+b$

$=a^{5} b-a^{5} b-a^{3} b+2 a^{3} b-a^{3} b+a^{2} b-2 a^{2} b+a^{2} b+b$

$=b$