Does perfect square imply perfect square polynomial?

Algebra Level 2

True or false:

Consider a quadratic expression \(f(x) = x^2 + bx + c\), where \(b\) and \(c\) are rational numbers. If for all rational numbers \(r\), \(f(r) \) is the square of a rational number, then \(f(x) \) must be the square of a linear polynomial.

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