The correct response is: No, because there would have to be two recessive alleles for the foal to have a red coat.
To explain, for the foal to have a red coat, it would need to inherit two recessive alleles (ee). Since the stallion is homozygous dominant (EE), he can only pass on the dominant allele (E). The mare, being heterozygous (Ee), can pass on either the dominant allele (E) or the recessive allele (e). However, even if the foal inherits the recessive allele (e) from the mare, it will still have at least one dominant allele (E) from the stallion, resulting in a coat color that expresses the dominant trait (black). Thus, a red-coated foal cannot be produced from this cross.