What will be the quotient and remainder be on division of \[a{x^2} + bx + c\] by \[p{x^3} + q{x^2} + rx + s,p \ne 0\]?

Hint: Here the question is related to the algebraic expressions. We are applying the division arithmetic operation on the algebraic expressions, so we are dividing the given algebraic expressions and then we are determining the quotient and remainder and hence it will be the required result.

Complete step-by-step answer:
The given terms are algebraic expressions of different degrees. The algebraic expression is defined as the combination of the constants and variables.
Now we have to divide these two given algebraic expressions.
On considering the given question
Here we have to divide \[a{x^2} + bx + c\] by \[p{x^3} + q{x^2} + rx + s\]. The dividend is \[a{x^2} + bx + c\] and the divisor is \[p{x^3} + q{x^2} + rx + s\]
The degree of the algebraic expression \[a{x^2} + bx + c\] is 2 and the degree of the algebraic expression \[p{x^3} + q{x^2} + rx + s\] is 3. Strictly it is mentioned that if p is not equal to zero then it will be a large or greater algebraic expression.
The degree of the dividend algebraic expression is lesser than the degree of the divisor algebraic expression, so it is highly impossible to divide.
So we can’t divide it.
So now we multiply the divisor algebraic expression by zero and then we are simplifying it
Therefore the quotient is 0 and remainder is \[a{x^2} + bx + c\]when we divide the \[a{x^2} + bx + c\] by \[p{x^3} + q{x^2} + rx + s,p \ne 0\]

Note: When we are dividing the algebraic expressions then the dividend and divisor algebraic expressions should have the same degree or the degree of the divisor algebraic expression should be less than the degree of the dividend algebraic expression. This rule is only for the division.