#### Answer

vertical asymptote $x=-2$
horizontal asymptote $y=1$
no slant asymptote.
See graph.

#### Work Step by Step

Step 1. Factor the numerator and the denominator; we have $h(x)=\frac{x^2+4x+3}{(x+2)^2}=\frac{(x+3)(x+1)}{(x+2)^2}$ and there are no common factors.
Step 2. The vertical asymptote can be found as $x=-2$
Step 3. The horizontal asymptote can be found when $x\pm\infty$ and we get $y=1$
Step 4. A slant asymptote exists if the numerator is one order higher than the denominator; for the given rational function, there is no slant asymptote.
Step 5. See graph.