How do you divide #( x^3 + 8x^2 + 19x + 12 )/(x^2+2)#?
I like to long divide just the coefficients, not forgetting to include
The process is similar to long division of numbers.
Write the dividend under the bar and the divisor to the left of the bar.
Write out the quotient one term at a time, choosing it to cause the leading terms of the running remainder to match.
Subtract the product of the divisor and the term of the quotient from the running remainder and bring down the next term from the dividend alongside it.
Repeat until the running remainder is shorter than the divisor.
In our case we get a quotient
#(x^3+8x^2+19x+12)/(x^2+2) = x+8 + (17x-4)/(x^2+2)#