(Originais Teachy 2023) - Question Medium of Mathematics

Let P(x) be a polynomial of degree 4 and Q(x) a polynomial of degree 2. When the polynomial P(x) is divided by Q(x), the remainder of the division is a polynomial of degree at most 1. Explain why this occurs, using the properties of polynomials to justify your answer. In addition, mathematically demonstrate that the degree of the polynomial resulting from the division of P(x) by Q(x) is at most 2, considering that the degree of the quotient polynomial is the difference between the degrees of the polynomials P(x) and Q(x) and that the degree of the remainder polynomial is less than the degree of the divisor polynomial Q(x).