Such division is generally performed using synthetic division or long division. The synthetic division tableau is shown below. The result is
(4x² + x - 3) ÷ (x -1) = 4x +5 +2/(x-1)
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Long division is performed exactly the same way as with numbers. For each quotient term, compute the ratio of the highest degree dividend term and the highest degree divisor term. That is, the first quotient term is (4x²)/(x) = 4x. Then, as with numerical long division, you multiply this by the divisor and subtract from the dividend to get the new dividend. Repeat this sequence of steps until the degree of the dividend is less than that of the divisor. That remaining dividend is the remainder. As with numerical long division, you can write the remainder as a fraction added to the quotient (see above).