The function f satisfies $f\left(x\right)+f(2x+y)+5xy=f(3x-y)+2x^2+1$
for all real numbers $x,y.$ Determine the value of $f\left(10\right).$

 

-49 Setting  $x=10$  and  $y=5$ gives  $f\left(10\right)+f\left(25\right)+250=f\left(25\right)+200+1$, from which we get  $f\left(10\right)=-49$.
By putting $y=\frac{x}{2}$, we see that the function is  $f\left(x\right)=-\frac{1}{2}{x}^2+1$, and it can be checked that this function indeed satisfies the given equation.