2n2+10n+8=2(n+1)(n+4). Multiply through by this:
3=n(n+4)+2(n+4),
3=n2+4n+2n+8=n2+6n+8,
n2+6n+5=0=(n+1)(n+5), and n=-1 or -5.
However, n=-1 would make all the denominators zero. So this solution must be rejected, because the fractions cannot be defined. The only solution is n=-5.
CHECK
Plug in n=-5:
3/8=(-5)/(-8)+1/(-4)=5/8-1/4, which is true, so n=-5 is the confirmed solution.