This was more than we could bear, so we decided(谓)to spend our last day(宾补?) exploring the countryside.(宾)
to spend our last day exploring the countryside.整个不定式短语作为整体作decided的宾语。
exploring the countrside是to spend的状语,是不定式的内部结构,不参与句子层次的剖析。非谓语动词都可以带有我们的宾语和状语构成非谓语动词短语,然后整个非谓语动词短语在句子中作名词性、形容词性或副词性的成分。