The pilot and two passengers were on a cross-country flight. Upon arrival to the destination airport, the pilot entered the traffic pattern for landing. He reported that the downwind leg was too close to the runway, so he departed the pattern and re-entered for a more stable approach. During the landing he looked for birds and the displaced threshold; he flared the airplane too high, landed hard and bounced. The airplane landed hard a second time, before the pilot executed a go-around. The pilot then conducted a normal landing. Examination of the airplane revealed substantial damage to the firewall and damage to the propeller tips.