= Solution
{c}
For weight $x^2$, the moments are $\int_0^1x^{j+2}\,dx=1/(j+3)$. Monic orthogonalization gives $\phi_0=1$ and $\phi_1=x-(1/4)/(1/3)=x-3/4$.
For $\phi_2=x^2+ax+b$, orthogonality to $1,x$ requires
$$
\frac15+\frac a4+\frac b3=0,\qquad
\frac16+\frac a5+\frac b4=0.
$$
Solving gives $a=-4/3$, $b=2/5$. Hence
$$
\boxed{\phi_0=1,\quad\phi_1=x-\tfrac34,\quad
\phi_2=x^2-\tfrac43x+\tfrac25,}
$$
$$
\boxed{x_1=\frac{10-\sqrt{10}}{15},\qquad
x_2=\frac{10+\sqrt{10}}{15}.}
$$
These are the nodes for the <two-node Gaussian quadrature with quadratic weight>.
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