= Solution
{c}
For the weighted <inner product>, $(x^j,x^k)=1/(j+k+2)$. Start <Gram-Schmidt process> with the monomials and choose monic normalization:
$$
\phi_0=1,\qquad
\phi_1=x-\frac{(x,1)}{(1,1)}=x-\frac23.
$$
Its squared <norm> is $1/4-4/9+2/9=1/36$. Next $(x^2,1)/(1,1)=1/2$ and
$$
(x^2,\phi_1)=\frac15-\frac16=\frac1{30},
\qquad\frac{(x^2,\phi_1)}{\|\phi_1\|_2^2}=\frac65.
$$
Therefore
$$
\boxed{\phi_0=1,\qquad\phi_1=x-\frac23,\qquad
\phi_2=x^2-\frac65x+\frac3{10}.}
$$
Subtracting these <orthogonal> projections gives <orthogonality> by construction; independent nonzero rescalings would give equivalent unnormalized <orthogonal> <polynomials>.
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