Im trying to substract a set of functions
$$f(x)-g(x)-h(x)-i(x)$$
where $$ f(x)=\frac{1}{(x+3)(x+4)^2(x+5)^3} $$ $$ g(x)=\frac{-\frac{1}{2}}{(x+5)^3} $$ $$ h(x)=\frac{-1}{(x+4)^2} $$ $$ i(x)=\frac{\frac{1}{8}}{(x+3)} $$
I use the common denominator approach
$$\frac{1}{(x+3)(x+4)^2(x+5)^3}+\frac{\frac{1}{2}}{(x+5)^3}+\frac{1}{(x+4)^2}-\frac{\frac{1}{8}}{(x+3)}=$$
$$\frac{1+\frac{1}{2}(x+3)(x+4)^2+(x+3)(x+5)^3-\frac{1}{8}(x+4)^2(x+5)^3}{(x+3)(x+4)^2(x+5)^3}$$
but how can be reduced this?; it is supposed to yield
$$g(x)=\frac{20-3x-x^2}{8(x+4)(x+5)^2}$$
Thanks in advance Update> making the products $$\frac{1+\frac{1}{2}(x^3+11x^2+40x+48)+(x^4+18x^3+120x^2+350x+375)-\frac{1}{2}(x^5+23x^4+211x^3+965x^2+2200x+2000)}{(x+3)(x+4)^2(x+5)^3}$$ and multiplying by 8
$$\frac{8+(4x^3+44x^2+160x+192)+(4x^4+72x^3+480x^2+1400x+1500)-(x^5+23x^4+211x^3+965x^2+2200x+2000)}{8(x+3)(x+4)^2(x+5)^3}$$ $$\frac{-x^5-19x^4-135x^3-441x^2-640x-300}{8(x+3)(x+4)^2(x+5)^3}$$ but still, this doesnt look too much like the $g(x)$; how can simplify the expression?