I discovered him finished his homework.

这是错句。改为宾语从句结构：

I discovered he had finished his homework.

discover可后接现在分词或形容词作宾补，但不接过去分词作宾补，除非这个过去分词已经形容词化了。

你可以查词典，注意句型和例句。

同意曹老师和黎老师解答观点

discover后面只能接以下几种形式（不能接过去分词作宾补）

(1)discover+that/wh-从句

She discovered that they had escaped. 她发现他们已经逃跑了。

I discovered that I was pregnant.我发现自己怀孕了。

It was difficult for the inspectors to discover which documents were important.检查员们很难发现哪些文件是重要的。

His friends were shocked to discover how ill he was. 

朋友们发现他病得这么厉害都惊愕不已。

(2)discover sth

Haskell did not live to discover the deception.

哈斯克尔至死都没识破那个骗局。

Scientists around the world are working to discover a cure for AIDS. 

全世界的科学家都在努力寻找治疗艾滋病的方法

He is the first European to discover America.

他是第一个发现美洲大陆的欧洲人

(3)discover sb/sth doing / sb be discovered doing 

He was discovered hiding in a shed. 

=People discovered him hiding in a shed

人们发现他原来藏在棚屋里。 

(4)discover sb+adj./sb be discovered +adj

She was discovered dead at her home in Leeds. 

=People discovered her dead at her home in Leeds

人们发现她死在利兹她的家里

网友给出句子应该作以下修改

(1)I discovered him finished his homework. 

应该改为：I discovered that he had finished his homework.

(2)I discovered him exhausted.

应该改为：I discovered that he was exhausted