If a quartic equation has solutions in the rationals, can the quartic plus a rational constant also have solutions in the rationals?

Question

Examples would be appreciated or methods to find such equations and constants.

Edit: The equation is assume to have 4 rational solutions, not necessarily distinct.

Answer 1

$$ x^4 - 65 x^2 + 64 $$ $$ x^4 - 65 x^2 + 784 $$

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$$ x^4 - 1105 x^2 + 17424 $$ $$ x^4 - 1105 x^2 + 82944 $$ $$ x^4 - 1105 x^2 + 138384 $$ $$ x^4 - 1105 x^2 + 304704 $$

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$$ x^4 - 32045 x^2 + 128164 $$ $$ x^4 - 32045 x^2 + 11437924 $$ $$ x^4 - 32045 x^2 + 63329764 $$ $$ x^4 - 32045 x^2 + 123698884 $$ $$ x^4 - 32045 x^2 + 145491844 $$ $$ x^4 - 32045 x^2 + 182304004 $$ $$ x^4 - 32045 x^2 + 239568484 $$ $$ x^4 - 32045 x^2 + 255424324 $$

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