Analyze \((x^2~+~3x~+~5)/(x~+~1)~=~x~+~2~+~3/(x~+~1)\). State the trend, remainder, correction, restriction, correction limit, asymptote, and multiplication identity.

Quotient-plus-remainder form separates a rational function into a polynomial trend and a proper-fraction correction. The correction shrinks toward zero far from the origin, so the graph approaches the polynomial part, while the original denominator restriction still remains. Multiplying divisor by quotient and adding the remainder is the built-in check that the rewrite represents the same numerator.