Higher Paper 2

\[\begin{aligned} &\text{Inside the bracket:}\quad (x+2)\div\frac{2x^2+x-6}{2x-3} =(x+2)\cdot\frac{2x-3}{2x^2+x-6}.\\ &\text{Factor }2x^2+x-6=(2x-3)(x+2).\\ &\Rightarrow (x+2)\cdot\frac{2x-3}{(2x-3)(x+2)}=1\quad\big(\,x\ne -2,\;x\ne \tfrac{3}{2}\,\big).\\[4pt] &\therefore\; \frac{12}{5x-10}+1=\frac{12}{5x-10}+\frac{5x-10}{5x-10} =\frac{12+5x-10}{5x-10}=\frac{5x+2}{5x-10},\quad x\ne 2.\\ &\text{Overall domain from original expression: } x\ne -2,\;\tfrac{3}{2},\;2. \end{aligned}\]