my problem was $2x^4 + x^3 -19x^2 + 18x + 5$ divided by $2x-5$
my answer in long division was : $x^3 + 3x^2 - 2x + 4$ and a remainder of $25$ my answer in synthetic division : $2x^3 + 11x^2 + 36x + 198$ and remainder of $995$
2026-04-05 17:58:42.1775411922
My answer in long division and synthetic division dont match
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Your result from long division is correct. Indeed, \begin{align*} 2x^4 + x^3 - 19x^2 + 18x + 5 & = 2x^4 - 5x^3 + 6x^3 - 15x^2 - 4x^2 + 10x + 8x - 20 + 25 \\ & = x^3(2x - 5) + 3x^2(2x - 5) - 2x(2x - 5) + 4(2x - 5) + 25 \\ & = (2x - 5)(x^3 + 3x^2 - 2x + 4) + 25, \end{align*} so $$ \frac{2x^4 + x^3 - 19x^2 + 18x + 5}{2x - 5} = x^3 + 3x^2 - 2x + 4 + \frac{25}{2x - 5}. $$