Does perfect square imply perfect square polynomial?

Algebra Level 2

True or false:

Consider a quadratic expression f(x)=x2+bx+cf(x) = x^2 + bx + c, where bb and cc are rational numbers. If for all rational numbers rr, f(r)f(r) is the square of a rational number, then f(x)f(x) must be the square of a linear polynomial.

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