Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and deg p(x) = deg q(x)

Degree of p(x) = degree of q(x)
The degree of dividend and quotient can be equal, if the divisor is a constant (degree 0) term. Therefore
Let p(x) = 3x – 6x + 5
Let g(x) = 3
Therefore, q(x) = x² – 2x + 1 and r(x) = 2

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Euclid’s division lemma can also be generalized to polynomials. For example, suppose we want to divide the polynomial x³ + 2x² – 3x + 4 by the polynomial x – 1. According to Euclid’s division lemma, there exist unique polynomials Q and R such that x³ + 2x² – 3x + 4 = (x – 1) Q + R, where the degree of R is less than the degree of x – 1. In this case, we can see that Q = x² + 3x – 4 and R = -x + 4 satisfy the conditions of the lemma, so the quotient is x² + 3x – 4 and the remainder is -x + 4 when x³ + 2x² – 3x + 4 is divided by x – 1.