Multiply $ \dfrac{x^2+3x+2}{x^2+3x} $ by $ \dfrac{x^2}{x+1} $ to get $ \dfrac{ x^3+2x^2 }{ x^2+3x } $.

Step 1: Factor numerators and denominators.

Step 2: Cancel common factors.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} \frac{x^2+3x+2}{x^2+3x} \cdot \frac{x^2}{x+1} & \xlongequal{\text{Step 1}} \frac{ \left( x+2 \right) \cdot \color{blue}{ \left( x+1 \right) } }{ x^2+3x } \cdot \frac{ x^2 }{ 1 \cdot \color{blue}{ \left( x+1 \right) } } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ x+2 }{ x^2+3x } \cdot \frac{ x^2 }{ 1 } \xlongequal{\text{Step 3}} \frac{ \left( x+2 \right) \cdot x^2 }{ \left( x^2+3x \right) \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ x^3+2x^2 }{ x^2+3x } \end{aligned} $$