### Instruction: Find $g(x)$, with terms in order of decreasing degree, if we have that \[8x^4-7x^2+8x-7+g(x)=x + 1.\].### Response: This problem asks me to find a function $g(x)$ that satisfies a given equation.
To do that, I need to isolate $g(x)$ on one side of the equation, and then simplify the other side as much as possible.
I can start by subtracting $x + 1$ from both sides, to get \[8x^4-7x^2+8x-7+g(x)-x-1=0.\]
Then, I can rearrange the terms on the left side and group them by their degree, to get \[8x^4-7x^2+7x-8+g(x)=0.\]
Now, I can subtract the polynomial part from both sides, to get \[g(x)=-8x^4+7x^2-7x+8.\]
This is the expression for $g(x)$, and it is already in order of decreasing degree.