This question clearly states that you will be investigating:

1. The influence of parental genes on offspring
2. The impact on both the genotype (genetic makeup) and phenotype (observable traits) of the offspring mice

You can propose a hypothesis based on Mendelian genetics principles to develop your experiment further. For example:
Hypothesis: If a single gene controls a specific trait and follows Mendelian inheritance patterns, breeding two mice with known genotypes will result in offspring with predictable genotypes and phenotypes based on segregation and independent assortment laws.
Next steps:

1. Select a specific trait to study (e.g., coat color, ear shape, or tail length)
2. Determine the mode of inheritance for the trait (dominant/recessive, complete/incomplete dominance, codominance, etc.)
3. Choose parental mice with known genotypes for the selected trait
4. Breed the parental mice and analyze the genotypes and phenotypes of the offspring
5. Compare the observed results to the expected results based on your hypothesis and Mendelian genetics principles

This experimental design will allow you to investigate how parental genes influence the inheritance of specific traits in their offspring and test the principles of Mendelian genetics in mice.