1. I think the promotion is actually going very well.
我认为促销进展得真的很不错。
还能这样说：
I think our promotion is very good.
I'm satisfied with our promotion.
谚语：
Think well before you act.
请三思而后行。

2. I wish us to succeed in our business transaction.
希望我们生意兴隆。
还能这样说：
I hope our transaction goes well.
Wish us a good luck in business.
应用解析：
hole-and-conner transactions 秘密的交易，偷偷摸摸的交易

3. We have made a very good start in our business.
我们在业务上有了良好的开端。
还能这样说：
Our business has a good beginning.
Our business has started successfully.
谚语：
Clumsy birds have to start flying early.
笨鸟先飞。

4. We hope for long term cooperative relationships with customers through our high quality products and competitive prices.
我们真诚希望凭借自己高质量的产品和有竞争力的价格与客户建立长久的业务联系。
还能这样说：
We hope we can maintain a close partnership with our customers by keeping high quality products and competitive prices.
We hope that the high quality products and competitive prices can bring us long term cooperation with the customers.
应用解析：
commercial partnership 商业合伙；
family partnership 家庭合伙；
joint-venture partnership 联营，合营