A function $f$ is continuous on the real line and satisfies $\displaystyle f(x) = 6x^2 - 6x + \int_0^1 (2x + t)\,f(t)\,dt \qquad \text{for every real } x.$ At first sight $f$ appears on both sides, so the equation looks circular. But inside the integral the variable is $t$, and $x$ is a constant there, so the integral can be split into $x$ times one number plus another number. Give names to those two numbers, write $f$ in terms of them, and find them from two simple equations. Each equation comes from integrating your formula for $f$ itself, so take care to integrate the right expression each time. What is the value of $f(2)$?