\[\left( x^{2} + 3x + 2 \right) \cdot \ *\ = x^{3} + 4x^{2} + 5x + 2\]
\[x^{3} + 4x^{2} + 5x + 2 =\]
\[= x^{3} + x^{2} + 3x^{2} + 3x + 2x + 2 =\]
\[= x^{2}(x + 1) + 3x(x + 1) + 2 \cdot (x + 1) =\]
\[= (x + 1)\left( x^{2} + 3x + 2 \right)\]
\[\left( x^{2} + 3x + 2 \right) \cdot \ *\ =\]
\[= \left( x^{2} + 3x + 2 \right)(x + 1)\]
\[\Longrightarrow *\ = x + 1.\]