Rewrite expressions of the form a(x)/b(x), where a and b are polynomials, in the form q(x)+r(x)/b(x), where q and r are polynomials and the degree of r is less than the degree of b.

Problem

Let a, left parenthesis, x, right parenthesis, equals, 5, x, start superscript, 3, end superscript, plus, 2, x, start superscript, 2, end superscript, plus, x, plus, 2, and b, left parenthesis, x, right parenthesis, equals, x, start superscript, 3, end superscript, plus, x, plus, 1.
When dividing a by b, we can find the unique quotient polynomial q and remainder polynomial r that satisfy the following equation:
start fraction, a, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fraction, equals, q, left parenthesis, x, right parenthesis, plus, start fraction, r, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fraction,
where the degree of r, left parenthesis, x, right parenthesis is less than the degree of b, left parenthesis, x, right parenthesis.
What is the quotient, q, left parenthesis, x, right parenthesis?
q, left parenthesis, x, right parenthesis, equals
What is the remainder, r, left parenthesis, x, right parenthesis?
r, left parenthesis, x, right parenthesis, equals