Upper bound for the error of the gaussian quadrature$\int_{0}^{1} \log (1+\operatorname{sin} x) \mathrm{d} x$

Question

I'm given the integral $\mathrm{I}=\int_{0}^{1} \log (1+\operatorname{sin} x) \mathrm{d} x$. Through the formula of Gaussian quadrature for 3 points, I can find an approximation to this integral. The question I'm asked is to show that $$ 2 \sup _{x \in[0,1]}|\log (1+\operatorname{sin} x)-P(x)| $$ is an upper bound for the error of the Gaussian quadrature, knowing that $P(x)$ is the polynomial with degree $\leq 5$ that approximates $\log (1+\operatorname{sin} x)$ through least squares. I don't even know where to start... Should I try to compute $P(x)$ and find the supremum? How would I show that it is an upper bound for the error?