10. \( \left(r^{2}+14 r+38\right) \div(r+8) \)

To solve \( \left(r^{2}+14 r+38\right) \div(r+8) \), you can use polynomial long division. First, divide the leading term of the numerator (\(r^2\)) by the leading term of the denominator (\(r\)), which gives \(r\). Multiply \(r\) by \(r + 8\) to get \(r^2 + 8r\), and subtract this from the original polynomial: \[ (r^2 + 14r + 38) - (r^2 + 8r) = 6r + 38. \] Next, divide the new leading term \(6r\) by \(r\) to get \(6\). Multiply \(6\) by \(r + 8\) to yield \(6r + 48\), and subtract again: \[ (6r + 38) - (6r + 48) = -10. \] Thus, the quotient is \(r + 6\) with a remainder of \(-10\). So, the result of the division can be expressed as: \[ r + 6 - \frac{10}{r + 8}. \] And there you have it! This polynomial division simplifies the expression nicely!