Answer
True
Work Step by Step
The given synthetic division is:-
$\begin{matrix}
&-- &-- &--&--& \\
-2) &5&3&2&1& & \\
& &-10 &14 &-32 && \\
& -- & -- & --& -- && \\
& 5 & -7& 16 &-31 & \\
\end{matrix}$
We have $c=2$, so the divisor is $x+2$.
Use the coefficients on the right side as dividend in descending powers of $x$.
The dividend is $5x^3+3x^2+2x+1$
The remainder is $-31$.
The Quotient is $5x^2-7x+16$
We can write
$\dfrac{\text{Dividend}}{\text{Divisor}}=\text{Quotient} + \dfrac{\text{Remainder}}{\text{Divisor}}$
Substituting all the values yields:
$=\dfrac{5x^3+3x^2+2x+1}{x+2}=5x^2-7x+16 + \dfrac{-31}{x+2}$
Hence, the given statement is true.
