Factor the expression completely: \[ x^{3}+3 x^{2}+2 x \]

To factor the expression \( x^{3}+3 x^{2}+2 x \) completely, we first factor out the greatest common factor, which is \( x \): \[ x(x^{2} + 3x + 2) \] Next, we need to factor the quadratic expression \( x^{2} + 3x + 2 \). We look for two numbers that multiply to \( 2 \) (the constant term) and add to \( 3 \) (the coefficient of \( x \)). The numbers are \( 1 \) and \( 2 \). Thus, we can write: \[ x^{2} + 3x + 2 = (x + 1)(x + 2) \] Now substituting this back into our expression, we get: \[ x(x + 1)(x + 2) \] So, the completely factored form of the expression \( x^{3}+3 x^{2}+2 x \) is: \[ x(x + 1)(x + 2) \]