Mechanical Integration

Taking $q(x)=1$, $q(x)=x$ and $q(x)=x^2$ in equation (1) and working out the integral (easy!) will give you three equations which you can solve to find $\lambda_1,\ \lambda_2$ and $\lambda_3$.

The key to showing that the same formula works for other polynomials is to show that if it works for $1,\ x$ and $x^2$ it works for any linear combination of them and so for any quadratic polynomial.

Finally you can go on to check out the formula for $x^3,\ x^4$ and $x^5$.