Find all quadratic polynomials that are orthogonal to the function $e^x$ with respect to the $L^2$ inner product on the interval $[0,1]$. 
Example: Find all quadratic polynomials that are orthogonal to the function $e^x$ with respect to the $L^2$ inner product on the interval $[0,1]$.
Solution: $p(x)=a((e-1)x-1)+b(x^2-(e-2)x)$ for any $(a,b\in\mathbb{R})$.

The textbook didn't give any further explanations or steps.  Can someone please help me understand how to get this result??

Quadratic polynomials are of the form $rx^2+sx+t$ for real numbers $r,s,t$. The inner product of one of them with $e^x$ is
$$
\int_0^1 (rx^2+sx+t)e^x \, dx = (e-2) r + s + (e-1) t.
$$
You're trying to find the values of $(r,s,t)$ for which the right-hand side equals $0$ (that's what orthogonal means). We've now reduced it to a linear algebra problem, which you should have techniques to solve.
Note that all we did at every step along the way was just write out the definitions of the terms in the question (and, at one step, a calculation), until we reduced the problem to solving a (system of) linear equation(s). This is a fruitful way to approach many linear algebra questions.
