Divide:

(i) 5m3 − 30m2 + 45m by 5m

$\left(5 m^{3}-30 m^{2}+45 m\right) \div 5 m$

$\Rightarrow \frac{5 m^{3}}{5 m}-\frac{30 m^{2}}{5 m}+\frac{45 m}{5 m}$

$\Rightarrow m^{2}-6 m+9$

Therefore, the quotient is m2 − 6m + 9

(ii) 8x2y2 − 6xy2 + 10x2y3 by 2xy

$\left(8 x^{2} y^{2}-6 x y^{2}+10 x^{2} y^{3}\right) \div 2 x y$

$\Rightarrow \frac{8 x^{2} y^{2}}{2 x y}-\frac{6 x y^{2}}{2 x y}+\frac{10 x^{2} y^{3}}{2 x y}$

$\Rightarrow 4 x y-3 y+5 x y^{2}$

Therefore, the quotient is 4xy − 3y + 5xy2.

(iii) 9x2y − 6xy + 12xy2 by − 3xy

$\left(9 x^{2} y-6 x y+12 x y^{2}\right) \div-3 x y$

$\Rightarrow \frac{9 x^{2} y}{-3 x y}-\frac{6 x y}{-3 x y}+\frac{12 x y^{2}}{-3 x y}$

$\Rightarrow-3 x+2-4 y$

Therefore, the quotient is −3x + 2 − 4y.

(iv)12x4 + 8x3 − 6x2 by − 2x2

$\left(12 x^{4}+8 x^{3}-6 x^{2}\right) \div-2 x^{22}-4 x+3^{2}$

$\Rightarrow \frac{12 x^{4}}{-2 x^{2}}+\frac{8 x^{3}}{-2 x^{2}}-\frac{6 x^{2}}{-2 x^{2}}$

$\Rightarrow-6 x$

Therefore the quotient is −6x2 − 4x + 3.