Simplify using Synthetic Division

Step 1
Write down the first coefficient without changes:
−3223−10−3
−3
2
3−10−3
2

Step 2
Multiply the entry in the left part of the table by the last entry in the result row (under the horizontal line).
Add the obtained result to the next coefficient of the dividend, and write down the sum.
−3223(−3)⋅2=−63+(−6)=−3−10−3
−
3
2
3
−10−3
(
−
3
)
⋅
2
=
−
6
2
3
+
(
−
6
)
=
−
3

Step 3
Multiply the entry in the left part of the table by the last entry in the result row (under the horizontal line).
Add the obtained result to the next coefficient of the dividend, and write down the sum.
−3223−6−3−10(−3)⋅(−3)=9(−10)+9=−1−3
−
3
23
−
10
−3 −6
(
−
3
)
⋅
(
−
3
)
=
9
2
−
3
(
−
10
)
+
9
=
−
1

Step 4
Multiply the entry in the left part of the table by the last entry in the result row (under the horizontal line).
Add the obtained result to the next coefficient of the dividend, and write down the sum.
−3223−6−3−109−1−3(−3)⋅(−1)=3(−3)+3=0
−
3
23−10
−
3
−69
(
−
3
)
⋅
(
−
1
)
=
3
2−3
−
1
(
−
3
)
+
3
=
0

We have completed the table and have obtained the following resulting coefficients: 2,−3,−1,0
2
,
−
3
,
−
1
,
0
.
All the coefficients except the last one are the coefficients of the quotient, the last coefficient is the remainder.
Thus, the quotient is 2x2−3x−1
2
x
2
−
3
x
−
1
, and the remainder is 0
0
.
Therefore, 2x3+3x2−10x−3x+3=2x2−3x−1+0x+3=2x2−3x−1
2
x
3
+
3
x
2
−
10
x
−
3
x
+
3
=
2
x
2
−
3
x
−
1
+
0
x
+
3
=
2
x
2
−
3
x
−
1
2x3+3x2−10x−3x+3=2x2−3x−1+0x+3=2x2−3x−1