Perform the division. \[ \frac{p^{2}+3 p-33}{p+8} \]

Step 1 ：Set up the division. We are dividing \(p^{2}+3 p-33\) by \(p+8\).

Step 2 ：Divide the first term in the numerator by the first term in the denominator. \(p^{2}\) divided by \(p\) is \(p\). Write this above the division bar.

Step 3 ：Multiply the entire denominator by the term we just wrote above the division bar (\(p\)), and subtract this from the numerator. The result is \(-5p - 33\).

Step 4 ：Divide the first term of the new numerator (-5p) by the first term of the denominator (p). This gives -5. Write this above the division bar, next to the \(p\).

Step 5 ：Multiply the entire denominator by the term we just wrote above the division bar (-5), and subtract this from the new numerator. The result is 7.

Step 6 ：Now, the degree of the remaining term in the numerator (7) is less than the degree of the denominator (p+8), so we stop here. The 7 is the remainder.

Step 7 ：So, the division of \(\frac{p^{2}+3 p-33}{p+8}\) results in \(p-5\) with a remainder of 7. We can write this as \(p-5+\frac{7}{p+8}\).

Step 8 ：Check the result: If we multiply \(p-5+\frac{7}{p+8}\) by \(p+8\), we should get back to the original numerator \(p^{2}+3 p-33\).

Step 9 ：The result is \(p^{2}+3p-33\), which confirms that the division is correct.

Step 10 ：\(\boxed{p-5+\frac{7}{p+8}}\) is the final answer.