Learn Russian: Вчера мы предложили хорошую работу нашей старшей племяннице. - Yesterday we offered a good job to our older niece.

The verb предложить (to offer) takes a direct object in the Accusative case for the thing being offered, and an indirect object in the Dative case for the person receiving the offer. Here, the niece is the recipient, so the noun and its describing words take Dative endings.

What are the dictionary forms of нашей старшей and how do they get their Dative endings?

Their dictionary (Nominative) forms are наша (our) and старшая (older). In the feminine Dative case, the pronoun наша changes to нашей. The adjective старшая takes the soft ending -ей instead of the hard ending -ой because of a spelling rule: unstressed endings after the hushing consonant ш are spelled with an е.

Why is it хорошую работу rather than хорошая работа?

This is a quick reminder of the Accusative case, which is used for the direct object of a sentence. Since the job is the thing being offered, the feminine noun работа changes its ending to -у, and the adjective хорошая changes to -ую.

What is the infinitive form of предложили?

The infinitive is предложить (to offer). It is a perfective verb, which is used here to indicate a single, successfully completed action that happened yesterday.