factor $x^{3}+3x^{2}+2x$ completely.\n$x(x^{2}+3x + 2)$\n$x(x + 1)(x + 2)$\n$x(x - 1)(x - 2)$

Answer:

B. $x(x + 1)(x + 2)$

Explanation:

Step1: Factor out the GCF

First, find the greatest - common factor of the terms $x^{3}$, $3x^{2}$, and $2x$. The GCF of $x^{3}$, $3x^{2}$, and $2x$ is $x$. So, $x^{3}+3x^{2}+2x=x(x^{2}+3x + 2)$.

Step2: Factor the quadratic

Next, factor the quadratic expression $x^{2}+3x + 2$. We need to find two numbers that multiply to $2$ (the constant term) and add up to $3$ (the coefficient of the $x$ - term). The numbers are $1$ and $2$ since $1\times2 = 2$ and $1 + 2=3$. So, $x^{2}+3x + 2=(x + 1)(x + 2)$.

Step3: Write the complete factorization

Combining the two steps, we get $x^{3}+3x^{2}+2x=x(x + 1)(x + 2)$.