I will do the part with all three points and you can do the other with two points.
We are given that $f(x) = e^{2x} - x$, $x_0 = 1$, $x_1 = 1.25$, and $x_2 = 1.6$.
We are asked to construct the interpolation polynomial of degree at most two to approximate $f(1.4)$, and find an error bound for the approximation.
You stated that you know how to find the interpolating polynomial, so we get:
$$P_2(x) = 26.8534 x^2-42.2465 x+21.7821$$
The formula for the error bound is given by:
$$E_n(x) = {f^{n+1}(\xi(x)) \over (n+1)!} \times (x-x_0)(x-x_1)...(x-x_n)$$
Since we do not know where $\xi(x)$ is, we will find each error over the range and multiply those together, so we have:
$$\max_{(x, 1, 1.6)} |f'''(x)| = \max_{(x, 1, 1.6)} 8 e^{2 x} = 196.26$$
Next, we need to find:
$$\max_{(x, 1, 1.6)} |(x-1)(x-1.25)(x-1.6)| = 0.00754624$$
Thus we have an error bound of:
$$E_2(x) = \dfrac{196.26}{6} \times 0.00754624 \le 0.246838$$
If we compute the actual error, we have:
$$\mbox{Actual Error}~ = |f(1.4) - P_2(1.4)| = |15.0446 - 15.2698| = 0.2252$$
Best Answer
Let $(x_0;y_0)\dots. (x_n;y_n)$ be a set of $n+1$ data points one wants to interpolate. Lagrange interpolation gives the way to build the only one polynomial $L(x)$ of degree $d\leq n$ with : $$L(x_i) = y_i.$$ By using Lagrange interpolation one has : $$L(x) = \sum_{j = 0}^n y_j l_j(x),\quad l_j(x)=\prod_{i = 0, i\neq j}^n \frac{x-x_i}{x_j-x_i}.$$ $l_j(x)$ is a polynomial of degree $n$ with $l_j(x_i)=\delta_{ij}$. It is easy to prove the uniqueness of $L(x)$ with linear algebra.
Let $Q(x)$ be another polynomial of degree $d'\leq n$ verifying : $$Q(x_i) = y_i,$$ then $L-Q$ is a polynomial of degree $d'' \leq n$ vanishing in $n+1$ points. This is possible iff : $$L = Q.$$ So if the function you want to interpolate is a polynomial of degree $\leq n$ then $L(x)$ is exactly this function and the error generated by the interpolation method vanishes and the formula you give for the interpolation error is still correct. In this case, the only error term you may take into account is the numerical error generated by your computer. This is why this method is widely used to interpolate functions behaving like polynomials or more precisely like $exp(x)$.