Answer:
Me gusta; le gustan
Step-by-step explanation:
For the first sentence, the reason why it's "Me gusta" is that the sentence includes "Soy" meaning "I am" in Spanish, therefore telling us that Patricia is refering about herself. But when it comes to using the verb "gustar," we must use indirect object pronouns (me, them, him, her, us). Since she's talking about herself, she would say "me gusta." The reason for why it's "gusta" instead of "gustan" is that the verb "cocinar" is singular.
When it comes to the second sentence, the answer is "le gustan" because le means "him, her, or you." The mom would be "le" because it's refering about the mom. And the reason why it's "gustan" is that the subjects physically exist and touchable.
For example: Le gusta el collar y la pulsera (She likes the necklace and bracelet)
Both the necklace and bracelet are physically existing, and we can easily grab them, touch them, hold them, etc.
Another example: Le gusta la lealtad y la fuerza (She likes the loyalty and the strength)
Strength and loyalty don’t physically exist, and you can’t easily touch them because they’re not touchable.