前半部分是定语从句还是同位语从句？The reason why he was dismissed is that hewas careless.

应该是定语从句
你已经在why he was dismissed 处打括号了，省略不看，这个句子的意思仍然通顺。原因是他太粗心了。
reason这个先行词一定要用why来引导，语法书有的，你可以去查一查，加入why he was dismissed这个后置定语，意思是他被解雇的原因是因为他太粗心了。

同位语。。。。。

这句话中既有定语从句又有宾语从句
The reason is that hewas careless里的that是引导宾语从句的
The reason why he was dismissed 这是定语从句，也可以说是原因状语从句。
why he was dismissed 并不是reason的内容，而是 he was careless，why he was dismissed 是对reason的限制。故而是定语从句。

定语从句
先行词reason 去掉括号里的从句句子结构依然完整
原因（为什么他被解雇的）是他太粗心了。

定语从句