\(\sqrt[3]{x+1}+\sqrt[3]{x+2}=\sqrt[3]{-x-3}\)
\(\Leftrightarrow\left(\sqrt[3]{x+1}+\sqrt[3]{x+2}\right)^3=-x-3\)
\(\Leftrightarrow x+1+x+2+3\left(\sqrt[3]{x+1}\right)^2\sqrt[3]{x+2}+3\sqrt[3]{x+1}.\left(\sqrt[3]{x+2}\right)^2\text{}\text{ }=-x-3\)
\(\Leftrightarrow3x+6+3\left(\sqrt[3]{x+1}\right)^2\sqrt[3]{x+2}+3\sqrt[3]{x+1}.\left(\sqrt[3]{x+2}\right)^2\text{}\text{ }=0\)
\(\Leftrightarrow x+2+\left(\sqrt[3]{x+1}\right)^2\sqrt[3]{x+2}+\sqrt[3]{x+1}.\left(\sqrt[3]{x+2}\right)^2\text{}\text{ }=0\)
\(\Leftrightarrow x+2+\sqrt[3]{x+1}\sqrt[3]{x+2}\left(\sqrt[3]{x+1}+\sqrt[3]{x+2}\right)\text{}=0\)
\(\Leftrightarrow\sqrt[3]{x+1}.\sqrt[3]{x+2}.\left(-\sqrt[3]{x+3}\right)\text{}=-x-2\)
\(\Leftrightarrow\sqrt[3]{x+1}.\sqrt[3]{x+2}.\sqrt[3]{x+3}\text{}=x+2\)
\(\Leftrightarrow\left(x+1\right).\left(x+2\right).\left(x+3\right)\text{}-\left(x+2\right)^3=0\)
\(\Leftrightarrow\left(x+2\right)\left(\left(x+1\right)\left(x+3\right)\text{}-\left(x+2\right)^2\right)=0\)
\(\Leftrightarrow\left(x+2\right)\left(x^2+4x+3\text{}-x^2-4x-4\right)=0\)
\(\Leftrightarrow-\left(x+2\right)=0\Leftrightarrow x=-2\)
