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

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Solution

We divide $3 x^{3} + 4 x^{2} + x + 7$ by $(x^{2} + 1)$ as shown in the above image:

From the division, we observe that the quotient is $3 x + 4$ and the remainder is $- 2 x + 3$ .

Now we will apply the division algorithm to check the answer as follows:
$\begin{matrix} Dividend & = Divisor \times Quotient + Remainder 3 x^{3} + 4 x^{2} + x + 7 & = [(x^{2} + 1) (3 x + 4)] - 2 x + 3 = (3 x^{3} + 4 x^{2} + 3 x + 4) - 2 x + 3 = 3 x^{3} + 4 x^{2} + 3 x - 2 x + 4 + 3 3 x^{3} + 4 x^{2} + x + 7 & = 3 x^{3} + 4 x^{2} + x + 7 [Which is true] \end{matrix}$

Hence verified

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