Divide using long division method and check the answer: $3{{x}^{2}}+4{{x}^{2}}+x+7$ by ${{x}^{2}}+1$.

Hint: For solving this type of question, we should know about the long division method. This method is used for dividing one large multi digit number by another large multi digit number. The formula for checking whether our answer is correct or not is given by the algorithm, Dividend = Divisor $\times $ Quotient + Remainder. So, we will first find the answer by using the long division method and then check if our answer is correct or not.

Complete step by step answer:
In the given question, we have to divide $3{{x}^{2}}+4{{x}^{2}}+x+7$ by ${{x}^{2}}+1$. So, by using the long division method for the same, we will get,
\[{{x}^{2}}+1\overset{3x+4}{\overline{\left){\begin{align}
  & 3{{x}^{3}}+4{{x}^{2}}+x+7 \\
 & \underline{3{{x}^{3}}\text{+ 0 + 3x }} \\
 & \text{ }4{{x}^{2}}-2x+7 \\
 & \text{ }\underline{\text{ }4{{x}^{2}}+\text{ }0\text{ }+4} \\
 & \text{ }-2x+3 \\
 & \text{ } \\
\end{align}}\right.}}\]
So, here, we have,
The divisor as ${{x}^{2}}+1$
The quotient as $3x+4$
And the remainder as $-2x+3$
We know that the division algorithm states that, Dividend = Divisor $\times $ Quotient + Remainder. So, we will apply that algorithm to the above division that we have performed using long division method to check the answer as follows,
Dividend = Divisor $\times $ Quotient + Remainder.
$\begin{align}
  & \Rightarrow 3{{x}^{3}}+4{{x}^{2}}+x+7=\left[ \left( {{x}^{2}}+1 \right)\left( 3x+4 \right) \right]-2x+3 \\
 & \Rightarrow 3{{x}^{3}}+4{{x}^{2}}+x+7=\left( 3{{x}^{3}}+4{{x}^{2}}+3x+4 \right)-2x+3 \\
 & \Rightarrow 3{{x}^{3}}+4{{x}^{2}}+x+7=3{{x}^{3}}+4{{x}^{2}}+x+4+3 \\
 & \Rightarrow 3{{x}^{3}}+4{{x}^{2}}+x+7=3{{x}^{3}}+4{{x}^{2}}+x+7 \\
\end{align}$

Therefore, we can conclude that our division is correct. And it is also clear that the method that we followed to get the answer is also correct.

Note: During the long division method, we should be very careful while dividing the ${{x}^{n}}$ values and putting the values under the same power of $x$ and after this you must change the sign. If the value has positive sign, then it will be negative and if the value has negative, then it will be positive.