Final answer:
To quickly find out if a wired-haired male dog carries the gene for smooth hair, he should be mated with a smooth-haired female. This test cross will reveal his genotype based on the puppies' phenotypes.
Step-by-step explanation:
In the case of the wired haired dogs, the wire hair trait is dominant, and the smooth hair trait is recessive. To determine most quickly whether the male wired-haired dog is a carrier of the smooth hair trait, it is advised to mate him with a female that has smooth hair (homozygous recessive). This type of mating test is generally known as a test cross. If the male is heterozygous (carries one allele for wire hair and one for smooth hair), mating with a smooth hair female could produce some puppies with smooth hair, indicating that the male carries the recessive allele. Conversely, if all puppies have wire hair, it can be inferred that the male is homozygous dominant for wire hair.