\(\sqrt{14+6\sqrt{5}}-\sqrt{\frac{\sqrt{5}-2}{\sqrt{5}+2}}=\sqrt{9+3.2\sqrt{5}+5}-\sqrt{\frac{\sqrt{5}-2}{\sqrt{5}+2}}=\sqrt{3^2+3.2\sqrt{5}+\left(\sqrt{5}\right)^2}-\sqrt{\frac{\sqrt{5}-2}{\sqrt{5}+2}}=\sqrt{\left(3+\sqrt{5}\right)^2}-\sqrt{\frac{\left(\sqrt{5}-2\right)^2}{\left(\sqrt{5}+2\right)\left(\sqrt{5}-2\right)}}=3+\sqrt{5}-\sqrt{\frac{\left(\sqrt{5}-2\right)^2}{5-2^2}}=3+\sqrt{5}-\sqrt{\left(\sqrt{5}-2\right)^2}=3+\sqrt{5}-\left(\sqrt{5}-2\right)\left(2=\sqrt{4}< \sqrt{5}\right)=3+\sqrt{5}-\sqrt{5}+2=5\)
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