ANSWER
Quotient:
[tex]q(x) = 4 {x}^{2} + 4x + 4[/tex]
Remainder:
[tex]r(x) = 2x + 6[/tex]
EXPLANATION
The given functions are
[tex]f(x) = 12 {x}^{4} + 20 {x}^{3} + 40 {x}^{2} + 30x + 25[/tex]
and
[tex]g(x) = 3 {x}^{2} + 2x + 5[/tex]
We want to determine the quotient and remainder when f(x) is divided by g(x).
We perform the long division as shown in the attachment.
The quotient is
[tex]q(x) = 4 {x}^{2} + 4x + 4[/tex]
and the remainder is
[tex]r(x) = 2x + 6[/tex]
