When to stop dividing in the long division of polynomials.

The four stages used to divide whole numbers—(divide, multiply, subtract, bring down the next term)—form the repeated process for the polynomial long division when a divisor has more than one term or if the divisor is a polynomial having more than one term.
The stages for dividing by a polynomial (or a lengthy division of polynomials) are listed below:
Sort the dividend and divisor's terms in decreasing powers of any variable.
Divide the dividend's first term by the divisor's first term. The first term of the quotient is the outcome.
Multiplying each phrase in the divisor by the quotient's first term Similar phrases should be put up beneath the dividend and the resulting product.
Subtract the dividend from the product.
To create a new dividend, lower the next term from the old dividend and write it next to the remaining terms.
Repeat this procedure using this new expression as the dividend until the remaining can no longer be split. When the degree of the remainder—the exponent with the largest value on a variable in the remainder—is lower than the degree of the divisor, this will happen.

$divisor=x^2+1,Remainder=x+5$
In this case, the residual has a degree of "one," which is lower than the divisor's degree of "two." At this point, division should cease since the remaining portion cannot be split further.