Answer: B. $x(11x^{2}+2x+7)$

Explanation:1. Identify the common factor: Look for the greatest common factor (GCF) among all terms in the expression $11x^{3}+2x^{2}+7x$. The terms are $11x^{3}$, $2x^{2}$, and $7x$. The numerical coefficients are 11, 2, and 7. Their GCF is 1. The variable parts are $x^{3}$, $x^{2}$, and $x$. The lowest power of x present in all terms is $x^{1}$ or $x$. So, the GCF of the expression is $x$.

2. Factor out the GCF: Divide each term in the expression by the GCF ($x$): $11x^{3} \div x = 11x^{3-1} = 11x^{2}$ $2x^{2} \div x = 2x^{2-1} = 2x^{1} = 2x$ $7x \div x = 7x^{1-1} = 7x^{0} = 7 \times 1 = 7$

3. Write the factored expression: The factored expression is the GCF multiplied by the sum of the results from step 2: $x(11x^{2}+2x+7)$

4. Check the options: Option [A]: $20(x^{3}+x^{2}+x) = 20x^{3}+20x^{2}+20x$. This is not equivalent. Option [B]: $x(11x^{2}+2x+7) = x \cdot 11x^{2} + x \cdot 2x + x \cdot 7 = 11x^{3}+2x^{2}+7x$. This is equivalent. Option [C]: $7x(11x^{3}+2x^{2}+1) = 77x^{4}+14x^{3}+7x$. This is not equivalent. Option [D]: $11x(x^{2}+2x+7) = 11x^{3}+22x^{2}+77x$. This is not equivalent.

Conclusion:The expression equivalent to $11x^{3}+2x^{2}+7x$ is $x(11x^{2}+2x+7)$.