There are no common factors between the numerator and denominator so this fraction will not reduce. But we can use long division. Long division with polynomials is similar to long division of numbers. You figure out many times the divisor will go into the first part of the dividend, multiply, subtract, bring down the next part of the dividend and repeat until there is no more of the dividend to bring down.h this division:
6b^2 + (-2b) + 8
__________________________
5b + 3 / 30b^3 + 8b^2 + 34b + 30
- (30b^3 + 18b^2)
---------------
-10b^2 + 34b
- (-10b^2 + (-6b))
--------------
40b + 30
- (40b + 24)
--------
6
We stop here because 5b, with its variable, cannot divide into 6 which has no variable. 6 is the remainder. And, if you remember from when you divided numbers, you make a fraction out the the remainder and the divisor. So:
Of course the "+ (-2b)" can be written as a subtraction: