Having been failed so many times in math exams, I was anxious about the subject.

这个句子的根本错误在于：不及物动词的fail当作及物动词使用。也就是说，它被错误地用了被动语态。

动词fail 无论使用什么时态或何种非谓语形式，主语是“人”时，都不能出现be failed 的任何形式。也就是说，以下表达均不成立：

I am failed in the math exam.

I am being failed in the math exam.

I was failed in the math exam.

I was being failed in the math exam.

I have been failed in the math exam.

I had been failed in the math exam.

I will be failed in the math exam.

I would be failed in the math exam.

Failed in the math exam, I...

Being failed in the math exam, I...

Having been failed in the math exam, I...

关于提问的句子，应去掉been，Having failed so many times in math exams 叫作现在分词的完成形式，它相当于一个发生在主句动作之前的原因状语从句：

Because I had failed so many times in math exams, I was anxious about the subject.