Use synthetic division to divide 2x^3 + 2x^2 + 2x - 1 by x - 1.

Step-by-step solution

1. 1

   Read off $c$ from the divisor. Synthetic division works with a divisor written as $x-c$. Here $x-1$ means $c=+1$ - the sign flips, and using $c=-1$ is the single most common error.

2. 2

   List the coefficients including any zeros. The cubic has every power present, so no zero placeholders are needed:

   $2,\ 2,\ 2,\ -1$

3. 3

   Run the bring-down / multiply / add cycle with $c=1$. Bring down $2$; multiply by $1$ and add to the next coefficient, repeatedly:

   $\begin{array}{r|rrrr}1 & 2 & 2 & 2 & -1\\ & & 2 & 4 & 6\\ \hline & 2 & 4 & 6 & 5\end{array}$

4. 4

   Read the answer off the bottom row. The last entry is the remainder; the others are the quotient coefficients, one degree lower than the dividend:

   $\text{quotient }=2x^{2}+4x+6,\qquad \text{remainder }=5$

5. 5

   Confirm the remainder with the remainder theorem. Dividing by $x-1$ must leave $f(1)$:

   $f(1)=2+2+2-1=5$

   which matches the bottom-right entry ✓ - an independent check that costs one line.

6. 6

   Verify the full identity by expanding.

   $(x-1)(2x^{2}+4x+6)+5=2x^{3}+4x^{2}+6x-2x^{2}-4x-6+5=2x^{3}+2x^{2}+2x-1$

   exactly the original polynomial ✓.