Step 1 :Given expression: $(\sqrt[3]{x+6}-2)\left(\sqrt[3]{(x+6)^{2}}-2 \sqrt[3]{x+6}+4\right)$
Step 2 :Expand the expression: $(\sqrt[3]{x+6}-2)(-2\sqrt[3]{x+6}+\sqrt[3]{(x+6)^{2}}+4)$
Step 3 :Final Answer: \(\boxed{(-2\sqrt[3]{x+6}+\sqrt[3]{(x+6)^{2}}+4)(\sqrt[3]{x+6}-2)}\)