Check whether $x + 1$ is a factor of ${x^3} + 3{x^2} + 3x + 1$ or not.

Hint: An expression \[x + a\] is a factor of a polynomial $p\left( x \right)$ only when $p\left( { - a} \right) = 0$, which means on putting $x = - a$ in the polynomial, zero must come out as result. Thus for $x + 1$, put $x = - 1$ in the polynomial given in the question and check whether it comes out to be 0 or not.

Complete step-by-step answer:
According to the question, the given polynomial is ${x^3} + 3{x^2} + 3x + 1$. Let this be $p\left( x \right)$.
So we have:
$ \Rightarrow p\left( x \right) = {x^3} + 3{x^2} + 3x + 1$.
We have to check whether $x + 1$ is a factor of $p\left( x \right)$ or not.
We know that an expression \[x + a\] is a factor of a polynomial $p\left( x \right)$ only when $p\left( { - a} \right) = 0$, which means on putting $x = - a$ in the polynomial, zero must come out as result.
In this case, we have $x + 1$ in place of \[x + a\]. So we will put $x = - 1$ instead of $x = - a$ in $p\left( x \right)$ and check if the result is 0 or not. By doing this, we’ll get:
$
   \Rightarrow p\left( { - 1} \right) = {\left( { - 1} \right)^3} + 3{\left( { - 1} \right)^2} + 3\left( { - 1} \right) + 1 \\
   \Rightarrow p\left( { - 1} \right) = - 1 + 3 - 3 - 1 \\
   \Rightarrow p\left( { - 1} \right) = 0
 $

Since 0 is coming out as a result, we can safely conclude that $x + 1$ is a factor of the polynomial ${x^3} + 3{x^2} + 3x + 1$.

Additional Information: If an expression \[x + a\] is a factor of a polynomial $p\left( x \right)$ then $x = - a$ is also a root of the polynomial $p\left( x \right)$. An ${n^{{\text{th}}}}$ degree polynomial can have at most $n$ real roots thus the polynomial can have maximum $n$ different factors. If we consider the polynomial mentioned above i.e. ${x^3} + 3{x^2} + 3x + 1$, this is a cubic polynomial. So it can have at most 3 different factors and 3 real roots.

Note: If on putting $x = - a$ in the polynomial $p\left( x \right)$, zero is not coming out as result then the residue is the remainder when the polynomial $p\left( x \right)$ is divided by \[x + a\]. This method is highly used in finding the remainder when a polynomial is divided by a linear, one variable expression.