Yes. If you add, subtract or multiply (but not if you divide) any two polynomials, you will get a polynomial.
8*7 = 56, plus the remainder.
I assume you meant x^4 + 5x^2 +10x + 12. The remainder is 28
+ (Add) - (Subtract) * (Multiply) / (Divide) % (Modulus aka. remainder)
The result is a polynomial q(x) whose order is one fewer than the order of p(x) and a remainder term of the form b/(x + a).
A fifth degree polynomial.
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The actual equation itself is the polynomial. There is no polynomial for it, and your question doesn't really make sense.
The remainder can only be one, so make it whatever number plus 1/2