Express the following as a partial fraction: (4x^2+12x+9) / (x^2+3x+2) .

We first notice that the degree of the numerator is the same as the degree of the denominator so we have an improper fraction which we need to separate into an integer and a proper fraction before we can express it in partial fractions.

We should try to see if there's a nice way to simplify the fraction before trying long division.

We have: (4x²+12x+9) / (x²+3x+2). We want to rearrange the numerator in such a way that it will cancel with the denominator to form an integer. We notice that if we factorise 4x²+12x into 4(x²+3x) it starts to look like the denominator. We just need +8 in the end instead of +9. So we rewrite 9 as 8+1 and we factorise to get:
4(x²+3x+2)+1 / (x²+3x+2) = 4 + 1/(x²+3x+2) = 4 + 1/(x+1)(x+2).

For now consider 1/(x+1)(x+2) = A/(x+1) +B/(x+2).

We need A(x+2)+B(x+1)=1. Equating coefficients: A+B=0 so A=-B and 2A+B=1 so 2A=1-B=1+A so A=1 and B=-1.
Thus, (4x²+12x+9) / (x²+3x+2) = 4 + 1/(x+1) -1/(x+2).

In the end if you have time, try to go backwards to double check if you find the initial improper fraction.