How do you factor by grouping $3{{x}^{3}}+2{{x}^{2}}+3x+2$ ?

Hint: Here in this question we have been asked to find the factors of the given cubic expression $3{{x}^{3}}+2{{x}^{2}}+3x+2$ . For doing that we will simplify this given expression and write it as $3{{x}^{3}}+2{{x}^{2}}+3x+2\Rightarrow {{x}^{2}}\left( 3x+2 \right)+\left( 3x+2 \right)$ . Later we will further simplify this and derive the factors.

Complete step by step answer:
Now considering from the question we have been asked to find the factors of the given cubic expression $3{{x}^{3}}+2{{x}^{2}}+3x+2$ .
For doing that we will simplify the given cubic expression by taking out the common terms and write it into a reduced for as $3{{x}^{3}}+2{{x}^{2}}+3x+2\Rightarrow {{x}^{2}}\left( 3x+2 \right)+\left( 3x+2 \right)$ .
Now this cubic expression can be further simplified by taking out more common terms. After further simplifying the cubic expression we will have $\Rightarrow \left( {{x}^{2}}+1 \right)\left( 3x+2 \right)$ .

Therefore we can conclude that the factors of the given cubic expression $3{{x}^{3}}+2{{x}^{2}}+3x+2$ are $\left( {{x}^{2}}+1 \right)\left( 3x+2 \right)$ .

Note: During the process of answering questions of this type we should be sure with the concepts that we apply. This is a very easy question and it can be answered accurately in a short span of time. This cubic expression can be further simplified by using the concept of complex numbers. The basic concept of complex numbers states that $i=\sqrt{-1}$ . Hence we can say that $i$ is a solution for the quadratic expression ${{x}^{2}}+1$. Therefore we can conclude that the factors of the given cubic expression $3{{x}^{3}}+2{{x}^{2}}+3x+2$ are $\left( x+i \right)\left( x-i \right)\left( 3x+2 \right)$ . If our question related concept is restricted to real numbers then we can end up as shown in the solution process if we have the liberty to use complex numbers then we answer as shown now.