why would eukaryotic DNA be incompatible with the process of protein synthesis in bacterial cells?

Eukaryotic DNA contains a lot of junk sequences called introns. During RNA transcription into mRNA, introns are cleaved from the RNA strand resulting in a coding strand of mRNA.

Subsequently, the mRNA is acted upon by the ribosomes to form the polypeptide.

The bacterial cell does not contain structures to cleave out the introns from the primary RNA transcript. Therefore the RNA translated by the bacterial cell would form a non functional protein.

> why would eukaryotic DNA be incompatible with the process of protein synthesis in bacterial cells?

a) Eukaryotic genes have a eukaryotic promoter sequence.  Prokaryotic transcription factors would not recognize that, and the mRNA would never be transcribed.
b) Eukaryotic genes have introns.  Prokaryotes would not properly snip these out from the mRNA transcript, so the polypeptide (protein) sequence the prokaryote made wouldn't be right.> how could eukaryotic DNA sequences be adapted to allow the bacterial cell to manufacture the required enzyme proteins?

Basically, take a mature eukaryotic mRNA transcript (one that has already had the introns snipped out).  Use reverse transcriptase to make a DNA copy of it.  Ligate this downstream from a prokaryotic promoter sequence.  Ligate that into a lysogenic bacterial pathogen virus.  Introduce virus into your bacterial culture.  Select.  Et voila.