\(A=\frac{x^2+5x+6+x\sqrt{9-x^2}}{3x-x^2+\left(x+2\right)\sqrt{9-x^2}}\)

\(=\frac{\left(x+2\right)\left(x+3\right)+x\sqrt{\left(3-x\right)\left(3+x\right)}}{x\left(3-x\right)+\left(x+2\right)\sqrt{\left(3-x\right)\left(3+x\right)}}\)

\(=\frac{\left(x+2\right)\left(x+3\right)+x\sqrt{\left(3-x\right)\left(3+x\right)}}{x\left(3-x\right)+\left(x+2\right)\sqrt{\left(3-x\right)\left(3+x\right)}}\)

\(=\frac{\sqrt{3+x}\left(\left(x+2\right)\sqrt{x+3}+x\sqrt{3-x}\right)}{\sqrt{3-x}\left(\left(x+2\right)\sqrt{x+3}+x\sqrt{3-x}\right)}\)

\(=\frac{\sqrt{3+x}}{\sqrt{3-x}}\)

\(B=\frac{x^2-5x+6+3\sqrt{x^2-6x+8}}{3x-12+\left(x-3\right)\sqrt{x^2-6x+8}}\)

\(=\frac{\left(x-3\right)\left(x-2\right)+3\sqrt{\left(x-4\right)\left(x-2\right)}}{3\left(x-4\right)+\left(x-3\right)\sqrt{\left(x-4\right)\left(x-2\right)}}\)

\(=\frac{\sqrt{x-2}\left(\left(x-3\right)\sqrt{x-2}+3\sqrt{x-4}\right)}{\sqrt{x-4}\left(3\sqrt{x-4}+\left(x-3\right)\sqrt{x-2}\right)}\)

\(=\frac{\sqrt{x-2}}{\sqrt{x-4}}\)

\(C=\frac{\sqrt{2\sqrt{4-x^2}}.\left(\sqrt{\left(2+x\right)^3}-\sqrt{\left(2-x\right)^3}\right)}{4+\sqrt{4-x^2}}\)

Đặt \(\hept{\begin{cases}\sqrt{2+x}=a\\\sqrt{2-x}=b\end{cases}\Rightarrow}a^2+b^2=4\)

\(\Rightarrow C=\frac{\sqrt{2ab}.\left(a^3-b^3\right)}{a^2+b^2+ab}=\frac{\sqrt{2ab}.\left(a-b\right)\left(a^2+b^2+ab\right)}{a^2+b^2+ab}\)

\(=\sqrt{2ab}.\left(a-b\right)=\sqrt{2\sqrt{4-x^2}}.\left(\sqrt{2+x}-\sqrt{2-x}\right)\)

Bạn muốn làm gì với biểu thức S???

\(=\frac{\left(x^2+5x+6\right)+x\sqrt{9-x^2}}{\left(3x-x^2\right)+\left(2+x\right)\sqrt{9-x^2}}\)

\(=\frac{\left(x+2\right)\left(3+x\right)+x\sqrt{\left(3+x\right)\left(3-x\right)}}{x\left(3-x\right)+\left(x+2\right)\sqrt{\left(3+x\right)\left(3-x\right)}}\) nhóm nhân tử chung

\(=\frac{\sqrt{3+x}\left(\left(x+2\right)\sqrt{3+x}+x\sqrt{3-x}\right)}{\sqrt{3-x}\left(x\sqrt{3-x}+\left(x+2\right)\sqrt{3+x}\right)}\)rồi rút gọn được

\(=\frac{\sqrt{3+x}}{\sqrt{3-x}}\)

Mình thấy biểu thức này không còn gì để rút gọn nữa đâu bạn.

Phân tích ra như vừa nãy 

Như sau :

\(D=\frac{\left(x+2\right)\left(x+3\right)+x\sqrt{\left(3-x\right)\left(3+x\right)}}{x\left(3-x\right)+\left(x+2\right)\sqrt{\left(3-x\right)\left(3+x\right)}}=\frac{\sqrt{x+3}\left[\left(x+2\right)\sqrt{x+3}+x\sqrt{3-x}\right]}{\sqrt{3-x}\left(x\sqrt{3-x}+\left(x+2\right)\sqrt{3+x}\right)}=\frac{\sqrt{x+3}}{\sqrt{3-x}}\)