\[ \begin{aligned} \frac{7a}{x^2-9}+\frac{5a}{x-3}-\frac{a}{x+3}-2 &=\frac{7a}{(x-3)(x+3)}+\frac{5a}{x-3}-\frac{a}{x+3}-2\\ &=\frac{7a+5a(x+3)-a(x-3)-2(x-3)(x+3)}{(x-3)(x+3)}\\ &=\frac{7a+5ax+15a-ax+3a-2(x^2-9)}{x^2-9}\\ &=\frac{25a+4ax-2x^2+18}{x^2-9} \end{aligned} \]

\[ \begin{aligned} \frac{x^2+y^2}{x^2-y^2}-\frac{x+y}{2x-2y}+1 &=\frac{x^2+y^2}{(x-y)(x+y)}-\frac{x+y}{2(x-y)}+1\\ &=\frac{2(x^2+y^2)-(x+y)^2+2(x-y)(x+y)}{2(x-y)(x+y)}\\ &=\frac{2x^2+2y^2-(x^2+2xy+y^2)+2(x^2-y^2)}{2(x^2-y^2)}\\ &=\frac{3x^2-2xy-y^2}{2(x^2-y^2)} \end{aligned} \]

\[ \begin{aligned} \frac{7}{2x-4}-\frac{3}{x+2}-\frac{12}{x^2-4}+x &=\frac{7}{2(x-2)}-\frac{3}{x+2}-\frac{12}{(x-2)(x+2)}+x\\ &=\frac{7(x+2)-6(x-2)-24+2x(x-2)(x+2)}{2(x-2)(x+2)}\\ &=\frac{7x+14-6x+12-24+2x(x^2-4)}{2(x^2-4)}\\ &=\frac{x+2+2x^3-8x}{2(x^2-4)}\\ &=\frac{2x^3-7x+2}{2(x^2-4)} \end{aligned} \]

\[ \begin{aligned} &\frac{7}{8a^2-18b^2}+\frac{1}{2a^2+3ab}-\frac{1}{4ab-6b^2}-2a\\ &=\frac{7}{2(4a^2-9b^2)}+\frac{1}{a(2a+3b)}-\frac{1}{2b(2a-3b)}-2a\\ &=\frac{7}{2(2a-3b)(2a+3b)}+\frac{1}{a(2a+3b)}-\frac{1}{2b(2a-3b)}-2a\\ &=\frac{7ab+2b(2a-3b)-a(2a+3b)-4a^2b(2a-3b)(2a+3b)}{2ab(2a-3b)(2a+3b)}\\ &=\frac{7ab+4ab-6b^2-2a^2-3ab-4a^2b(4a^2-9b^2)}{2ab(4a^2-9b^2)}\\ &=\frac{8ab-6b^2-2a^2-16a^4b+36a^2b^3}{2ab(4a^2-9b^2)} \end{aligned} \]

\[ \begin{aligned} \frac{2}{n+2}+\frac{n+3}{n^2-4}-\frac{3n+1}{n^2-4n+4} &=\frac{2}{n+2}+\frac{n+3}{(n-2)(n+2)}-\frac{3n+1}{(n-2)^2}\\ &=\frac{2(n-2)^2+(n+3)(n-2)-(3n+1)(n+2)}{(n+2)(n-2)^2}\\ &=\frac{-14n}{(n+2)(n-2)^2} \end{aligned} \]

\[ \begin{aligned} \frac{3}{2m+6}-\frac{m-2}{m^2+6m+9} &=\frac{3}{2(m+3)}-\frac{m-2}{(m+3)^2}\\ &=\frac{3(m+3)-2(m-2)}{2(m+3)^2}\\ &=\frac{m+13}{2(m+3)^2} \end{aligned} \]

\[ \begin{aligned} \frac{1+a}{a-3}-\frac{1-2a}{3+a}-\frac{a(1-a)}{9-a^2} &=\frac{1+a}{a-3}-\frac{1-2a}{a+3}+\frac{a(1-a)}{a^2-9}\\ &=\frac{(1+a)(a+3)-(1-2a)(a-3)+a(1-a)}{a^2-9}\\ &=\frac{2a^2-2a+6}{a^2-9} \end{aligned} \]

\[ \begin{aligned} \frac{1}{p-3}-\frac{3}{2p+6}-\frac{p}{2p^2-12p+18} &=\frac{1}{p-3}-\frac{3}{2(p+3)}-\frac{p}{2(p-3)^2}\\ &=\frac{2(p-3)(p+3)-3(p-3)^2-p(p+3)}{2(p-3)^2(p+3)}\\ &=\frac{-2p^2+15p-10}{2(p-3)^2(p+3)} \end{aligned} \]

\[ \begin{aligned} \frac{3x+2}{x^2-2x+1}-\frac{6}{x^2-1}-\frac{3x-2}{x^2+2x+1} &=\frac{3x+2}{(x-1)^2}-\frac{6}{(x-1)(x+1)}-\frac{3x-2}{(x+1)^2}\\ &=\frac{10x^2+10}{(x-1)^2(x+1)^2} \end{aligned} \]

\[ \begin{aligned} \frac{1}{a-b}-\frac{3ab}{a^3-b^3}-\frac{b-a}{a^2+ab+b^2} &=\frac{1}{a-b}-\frac{3ab}{(a-b)(a^2+ab+b^2)}+\frac{a-b}{a^2+ab+b^2}\\ &=\frac{a^2+ab+b^2-3ab+(a-b)^2}{(a-b)(a^2+ab+b^2)}\\ &=\frac{2(a-b)^2}{(a-b)(a^2+ab+b^2)}\\ &=\frac{2(a-b)}{a^2+ab+b^2} \end{aligned} \]

\[ \begin{aligned} \frac{1}{2+4m+2m^2}-\frac{1}{1-m^2}+\frac{1}{2-4m+2m^2} &=\frac{1}{2(1+m)^2}-\frac{1}{(1-m)(1+m)}+\frac{1}{2(1-m)^2}\\ &=\frac{(1-m)^2-2(1-m)(1+m)+(1+m)^2}{2(1+m)^2(1-m)^2}\\ &=\frac{4m^2}{2(1+m)^2(1-m)^2}\\ &=\frac{2m^2}{(1+m)^2(1-m)^2} \end{aligned} \]

\[ \begin{aligned} \frac{2}{3a+6}-\frac{a-2}{2a^2+4a}-\frac{2}{3a^2+12a+12}-\frac{4}{3a(a+2)^2} &=\frac{2}{3(a+2)}-\frac{a-2}{2a(a+2)}-\frac{2}{3(a+2)^2}-\frac{4}{3a(a+2)^2}\\ &=\frac{4a(a+2)-3(a+2)(a-2)-4a-8}{6a(a+2)^2}\\ &=\frac{4a^2+8a-3(a^2-4)-4a-8}{6a(a+2)^2}\\ &=\frac{a^2+4a+4}{6a(a+2)^2} =\frac{(a+2)^2}{6a(a+2)^2} =\frac{1}{6a} \end{aligned} \]