Final Answer
$4x^{2}+7x+19+\frac{28}{x-2}$
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Step-by-step Solution
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1
Let's divide the polynomial by $x-2$ using synthetic division (also known as Ruffini's rule). First, write all the coefficients of the polynomial in the numerator in descending order based on grade (putting a zero if a term doesn't exist). Then, take the first coefficient ($4$) and multiply it by the root of the denominator ($2$). Add the result to the second coefficient and multiply this by $2$ and so on
$\left|\begin{matrix}4 & -1 & 5 & -10 \\ & 8 & 14 & 38 \\ 4 & 7 & 19 & 28\end{matrix}\right|2$
2
In the last row appear the new coefficients of the polynomial. Use these coefficients to rewrite the new polynomial with a lower grade, and the remainder ($28$) divided by the divisor
$4x^{2}+7x+19+\frac{28}{x-2}$
Final Answer
$4x^{2}+7x+19+\frac{28}{x-2}$