Note that $x = -1$ is always a root of $x^3 + ax^2 + ax + 1 = 0,$ so we can factor out $x + 1,$ to get
\[(x + 1) (x^2 + (a - 1) x + 1) = 0.\]The quadratic factor has real roots if and only if its discriminant is nonnegative:
\[(a - 1)^2 - 4 \ge 0.\]This reduces to $a^2 - 2a - 3 \ge 0,$ which factors as $(a + 1)(a - 3) \ge 0.$  The smallest positive value that satisfies this inequality is $\boxed{3}.$
Question:
Let $a$ be a positive real number such that all the roots of
\[x^3 + ax^2 + ax + 1 = 0\]are real.  Find the smallest possible value of $a.$
Answer:
Let's think step by step.

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3
Let $a$ be a positive real number such that all the roots of
\[x^3 + ax^2 + ax + 1 = 0\]are real.  Find the smallest possible value of $a.$
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