Find the quotient and remainder using synthetic division: $\frac{x^{3}+7 x^{2}-51}{x+5}$ The quotient is The remainder is

Step 1 ：We are given the polynomial \(x^{3}+7 x^{2}-51\) and we are asked to divide it by \(x+5\).

Step 2 ：We will use synthetic division to find the quotient and remainder. Synthetic division is a shortcut method for dividing a polynomial by a linear polynomial of the form \(x - c\).

Step 3 ：In this case, the coefficients of the cubic polynomial are 1, 7, 0, -51 and the divisor is \(x + 5\), so \(c = -5\).

Step 4 ：Performing the synthetic division, we find that the quotient is \(x^{2}+2 x-10\) and the remainder is -1.

Step 5 ：Thus, we can express the original division as \(\frac{x^{3}+7 x^{2}-51}{x+5} = x^{2}+2 x-10 - \frac{1}{x+5}\).

Step 6 ：\(\boxed{x^{2}+2 x-10 - \frac{1}{x+5}}\) is the final answer.