Given a polynomial \( p(x) \), use synthetic division to divide by \( \boldsymbol{x}-a \) and obtain the quotient and the (nonzero) remainder. Write the results in the form quotient \( +\frac{\text { remainder }}{\text { divisor }} \) 4. \( \left(2 x^{3}+6 x^{2}-5 x+7\right) \div(x+3) \)

To perform synthetic division of the polynomial \( 2x^3 + 6x^2 - 5x + 7 \) by \( x + 3 \), we first need to use \( a = -3 \). We set up the synthetic division as follows: 1. Write down the coefficients of the polynomial: \( 2, 6, -5, 7 \). 2. Write \( -3 \) to the left. The process proceeds as follows: - Bring down the leading coefficient \( 2 \). \[ \begin{array}{r|rrrr} -3 & 2 & 6 & -5 & 7 \\ & & -6 & 0 & 15 \\ \hline & 2 & 0 & -5 & 22 \\ \end{array} \] - Multiply \( -3 \) by \( 2 \) to get \( -6 \) and add to \( 6 \) to get \( 0 \). - Multiply \( -3 \) by \( 0 \) to get \( 0 \) and add to \( -5 \) to get \( -5 \). - Multiply \( -3 \) by \( -5 \) to get \( 15 \) and add to \( 7 \) to get \( 22 \). The resulting coefficients \( 2, 0, -5 \) correspond to the quotient \( 2x^2 - 5 \) and the remainder is \( 22 \). Thus, we can express the division result as: \[ \text{quotient } = 2x^2 - 5 + \frac{22}{x + 3} \]