\(\sqrt{4+2\sqrt{3}}+\sqrt{4-2\sqrt{3}}-\dfrac{5}{\sqrt{3}-2\sqrt{2}}-\dfrac{5}{\sqrt{3}+\sqrt{8}}=\sqrt{3+2\sqrt{3}+1}+\sqrt{3-2\sqrt{3}+1}+\dfrac{5}{2\sqrt{2}-\sqrt{3}}-\dfrac{5\left(\sqrt{8}-\sqrt{3}\right)}{\left(\sqrt{8}+\sqrt{3}\right)\left(\sqrt{8}-\sqrt{3}\right)}=\sqrt{\left(\sqrt{3}+1\right)^2}+\sqrt{\left(\sqrt{3}-1\right)^2}+\dfrac{5\left(2\sqrt{2}+\sqrt{3}\right)}{\left(2\sqrt{2}-\sqrt{3}\right)\left(2\sqrt{2}+\sqrt{3}\right)}-\dfrac{5\left(\sqrt{8}-\sqrt{3}\right)}{8-3}=\left|\sqrt{3}+1\right|+\left|\sqrt{3}-1\right|+\dfrac{5\left(2\sqrt{2}+\sqrt{3}\right)}{8-3}-\left(\sqrt{8}-\sqrt{3}\right)=\sqrt{3}+1+\sqrt{3}-1+\left(2\sqrt{2}+\sqrt{3}\right)-\sqrt{8}+\sqrt{3}=4\sqrt{3}+2\sqrt{2}-2\sqrt{2}=4\sqrt{3}\)
