If dividing by $x+a$ or $x-2a$ gives the same remainder, what's $a$?

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The expressions \(px^4 - 5x + q\) and \(x^4 - 2x^3 - px^2 - qx - 8\) have a common factor \(x - 2\). Find the value of \(p\) and of \(q\).

The expression \(x^3 - x^2 - 2x - 3\) has the same remainder when divided by \(x + a\) and by \(x - 2a\). Find the non-zero values of \(a\) and the corresponding values of the remainder.