(a) Partial fractions $2/(x+2)+3/(x+3)$, then difference each term to get $-\dfrac{5x+14}{(x+2)(x+3)(x+4)}$. (b) $\sum PW=17429.76,\ \sum W=135$, so index $=129.11$.

Part (a) — $h=1$. Factor $x^{2}+5x+6=(x+2)(x+3)$ and split:

$\frac{5x+12}{(x+2)(x+3)}=\frac{A}{x+2}+\frac{B}{x+3}.$

$x=-2:\ 5(-2)+12=2=A\Rightarrow A=2.$ $x=-3:\ 5(-3)+12=-3=-B\Rightarrow B=3.$

So the function is $\dfrac{2}{x+2}+\dfrac{3}{x+3}.$ Using $\Delta\!\left[\dfrac{1}{x+a}\right]=\dfrac{1}{x+1+a}-\dfrac{1}{x+a}=-\dfrac{1}{(x+a)(x+a+1)}$:

$\Delta\!\left[\frac{2}{x+2}\right]=-\frac{2}{(x+2)(x+3)},\qquad \Delta\!\left[\frac{3}{x+3}\right]=-\frac{3}{(x+3)(x+4)}.$

Add:

$\Delta\!\left[\frac{5x+12}{x^{2}+5x+6}\right]=-\frac{2}{(x+2)(x+3)}-\frac{3}{(x+3)(x+4)}=-\frac{2(x+4)+3(x+2)}{(x+2)(x+3)(x+4)}=-\frac{5x+14}{(x+2)(x+3)(x+4)}.$

Part (b) — Cost-of-living index (family budget method): $P=\dfrac{p_1}{p_0}\times100$ (price relative), index $=\dfrac{\sum PW}{\sum W}.$

| Commodity | $p_0$(2007) | $p_1$(2011) | $P=\frac{p_1}{p_0}\times100$ | $W$ | $PW$ |

| --- | --- | --- | --- | --- | --- | …