Divide using synthetic division $$ \left(3 x^{2}+2 x-3\right) \div(x+3) $$

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We are asked to divide
$$
3x^2+2x-3
$$
by
$$
x+3.
$$

Since synthetic division is used when dividing by a linear factor, we first set the divisor equal to zero:
$$
x+3=0 \quad \Rightarrow \quad x=-3.
$$

The coefficients of the polynomial $$3x^2+2x-3$$ are $$3$$, $$2$$, and $$-3$$.

1. Write $$ -3 $$ to the left and list the coefficients:
   $$
   \begin{array}{c|ccc}
   -3 & 3 & 2 & -3 \
   \end{array}
   $$

2. Bring the first coefficient down:
   $$
   \begin{array}{c|ccc}
   -3 & 3 & 2 & -3 \
      &   &   &    \
      & 3 &   &    \
   \end{array}
   $$

3. Multiply the number just brought down by $$-3$$:
   $$
   3 \times (-3) = -9.
   $$
   Write $$-9$$ under the second coefficient:
   $$
   \begin{array}{c|ccc}
   -3 & 3 & 2 & -3 \
      &   & -9 &    \
      & 3 &   &    \
   \end{array}
   $$

4. Add the second coefficient and $$-9$$:
   $$
   2 + (-9) = -7.
   $$
   Write $$-7$$ as the next entry:
   $$
   \begin{array}{c|ccc}
   -3 & 3 & 2 & -3 \
      &   & -9 &    \
      & 3 & -7 &    \
   \end{array}
   $$

5. Multiply the newly obtained number $$-7$$ by $$-3$$:
   $$
   -7 \times (-3) = 21.
   $$
   Write $$21$$ under the third coefficient:
   $$
   \begin{array}{c|ccc}
   -3 & 3 & 2 & -3 \
      &   & -9 & 21 \
      & 3 & -7 &    \
   \end{array}
   $$

6. Add the third coefficient and $$21$$:
   $$
   -3 + 21 = 18.
   $$
   Place $$18$$ as the remainder:
   $$
   \begin{array}{c|ccc}
   -3 & 3 & 2 & -3 \
      &   & -9 & 21 \
      & 3 & -7 & 18 \
   \end{array}
   $$

The bottom row represents the coefficients of the quotient and the remainder. Since we started with a quadratic polynomial and divided by a linear polynomial, the quotient is a linear polynomial given by:
$$
3x - 7,
$$
and the remainder is $$18$$.

Thus, the result of the division is:
$$
\frac{3x^2+2x-3}{x+3}=3x-7+\frac{18}{x+3}.
$$
The division of $$3x^2 + 2x - 3$$ by $$x + 3$$ using synthetic division results in: $$ 3x - 7 \text{ with a remainder of } 18. $$ So, the final expression is: $$ 3x - 7 + \frac{18}{x + 3}. $$

Divide using synthetic division

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