jmc

Combining the fractions, we get $$\frac{2x+16}{(x+9)(x+7)}=\frac{2x+16}{(x+10)(x+6)}.$$Hence, $$(2x+16)(x+10)(x+6)=(2x+16)(x+9)(x+7),$$so $$2(x+8)[(x+10)(x+6)-(x+9)(x+7)]=2(x+8)(-3)=0.$$Thus, $x=\boxed{-8}.$