This sentence is still incorrect. The computer screen began flashing, and it would not stop. This may be the most common error in writing because it is the most common error in our speaking. We often use a plural pronoun to refer to a singular antecedent in our speech. This is acceptable when we speak, but it is not acceptable when we write.
Singular antecedents require singular pronouns.
The Supreme Court announced its decision today. Not The Supreme Court announced their decision today. Essential and non-essential clauses. Look up this entry in your AP Stylebook and study it carefully. Understand how an essential clause differs from a non-essential clause.