Exegesis:

kai ēlthon speusantes ‘they went with haste’; the participle speusantes qualifies the action of the main verb as happening with haste or quickly (cf. Revised Standard Version, New English Bible, Bible de Jérusalem, Zürcher Bibel).

speudō (also 19.5f) ‘to hurry,’ ‘to make haste,’ in Luke always as a participle qualifying the main verb.

kai aneuron ‘and they found.’

aneurō ‘to find out,’ implying a search as in Acts 21.4, cf. “they found their way to Mary and Joseph” (New English Bible).

tēn te Mariam kai ton Iōsēph kai to brephos keimenon en tē phatnē ‘Mary and Joseph and the child lying in the manger’; the series of connectives te … kai … kai admits the following interpretations: (1) ‘Mary, Joseph and the child,’ i.e. te is taken to be followed by more than one corresponding kai, cf. Moffatt; (2) ‘Mary and Joseph, and the child,’ i.e. te is connected with the first kai only and this connection expresses that Mary and Joseph are one group as compared with the child, cf. Revised Standard Version, Translator’s New Testament, Phillips, An American Translation, Zürcher Bibel; (3) the second kai is taken to be connect two co-ordinate clauses, cf. “they found … Mary and Joseph; and the baby was lying in the manger” (New English Bible). Of these interpretations (2) is the most probable because when te is followed by more than one kai usually te and the first kai connect concepts which are more closely related to each other than to what follows, cf. Lk. 12.45, Acts 1.8, Heb. 2.4; this is here supported by the fact that the participle keimenon ‘lying’ goes with brephos only.

Translation:

Found, or, because of the implied search, ‘they looked for and found,’ ‘in their search they saw’ (Cuyono).

Mary and Joseph, and the babe lying in a manger. To avoid the danger that the participle would be taken with the three persons mentioned, various ways are open, such as the use of different connectives, e.g. ‘Mary with Joseph, and the babe…’ (Bahasa Indonesia KB), ‘M. and J., more-over the babe…’ (Javanese, Balinese), or of a deictic element after ‘Joseph,’ giving to the next phrase a rather independent position (Sundanese); or the repetition of the verb, cf. ‘found M. with J., and found the babe lying…’ (Thai); or again, a repetition of ‘babe/child,’ cf. .’.. found M., J. and child; child lying…’ (Kituba).

Quoted with permission from Reiling, J. and Swellengrebel, J.L. A Handbook on the Gospel of Luke. (UBS Handbook Series). New York: UBS, 1971. For this and other handbooks for translators see here . Make sure to also consult the Handbook on the Gospel of Mark for parallel or similar verses.