How do you do these problems?
Let $f(x) = x^4-3x^2 + 2$ and $g(x) = 2x^4 - 6x^2 + 2x -1$. Let $a$ be a constant. What is the largest possible degree of $f(x) + a\cdot g(x)$?
Let $f(x) = x^4-3x^2 + 2$ and $g(x) = 2x^4 - 6x^2 + 2x -1$. Let $b$ be a constant. What is the smallest possible degree of the polynomial $f(x) + b\cdot g(x)$?
There is a polynomial which, when multiplied by $x^2 + 2x + 3$, gives $2x^5 + 3x^4 + 8x^3 + 8x^2 + 18x + 9$. What is that polynomial? (Give your answer as a polynomial in which the terms appear in order of decreasing degree. In other words, "$x^5 + x + 2$" is a valid answer, but "$2+x^5+x$" is not.)
The degree of a polynomial is determined by the largest power of x that occurs. Multiplying by a constant, no matter how big, makes no difference.
So, in "Let f(x) = x^4-3x^2 + 2 and g(x) = 2x^4 - 6x^2 + 2x -1. Let a be a constant. What is the largest possible degree of f(x) + a.g(x)?", the degree is 4.
Now, with that knowledge, see if you can do the other parts of your question.