solution: Expanding, we get
  \[\frac{(x + 5)(x + 2)}{x + 1} = \frac{x^2 + 7x + 10}{x + 1}.\]By long division,
  \[\frac{x^2 + 7x + 10}{x + 1} = x + 6 + \frac{4}{x + 1} = (x + 1) + \frac{4}{x + 1} + 5.\]By AM-GM,
  \[(x + 1) + \frac{4}{x + 1} \ge 2 \sqrt{(x + 1) \cdot \frac{4}{x + 1}} = 4,\]so $(x + 1) + \frac{4}{x + 1} + 5 \ge 9.$
  
  Equality occurs when $x = 1,$ so the minimum value is $\boxed{9}.$
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  Find the minimum value of
  \[\frac{(x + 5)(x + 2)}{x + 1}\]for $x > 0.$
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: 9
extra_info.question: Find the minimum value of
  \[\frac{(x + 5)(x + 2)}{x + 1}\]for $x > 0.$
extra_info.split: test