I'm not sure how to title this but here's the problem;
Define $a,b,c$ so that $P(x) + Q(x) = 0$ with every value of $x$.
$P(x) = 5x^2 - ax + 4 - (bx^2 - 7x + 3)$ and $Q(x) = 8x^2 + x + c$.
So you need to find values for $a, b$ and $c$ which will make the sum of $P(x)$ and $Q(x)$ equal to zero, no matter what you will input as the value of $x$.
I've tried simplifying the sum of the polynomials but that's how far I'll get. I can deduce the correct answer by trying different values but i don't think it's how you are supposed to solve this problem.
How you should mathematically approach this problem? Can you create an equation out of this?
In general what you want to do is group the like terms. What does $P(x)+Q(x)$ look like? $P(x)+Q(x)=5x^2-ax+4-bx^2+7x-3+8x^2+x+c$ now we can group the like terms to have that this is equal to: $(5-b+8)x^2+(-a+7+1)x+(4-3+c)$ we want this to equal zero, so in particular we need $(5-b+8)=0$, $(-a+7+1)=0$ and $(4-3+c)=0$.
The same strategy is how you would do this in general.