Answer
$\{1-2x+2x^2,\frac{2}{9}(16-5x-13x^2)\}$
Work Step by Step
According to Gram-Schmidt process, we obtain:
$v_1=p_1(x)=1-2x+2x^2\\
v_2=A_2-\frac{(A_2,v_1)}{||v_1||^2}v_1\\
=2-x-x^2-\frac{((2-x-x^2),(1-2x+2x^2))}{||1-2x+2x^2||^2}(1-2x+2x^2)\\
=2-x-x^2-\frac{2.1+(-1)(-2)+(-1).2}{1^2+(-2)^2+2^2} (1-2x+2x^2) \\
=2-x-x^2-\frac{2}{9}(1-2x+2x^2)\\
=\frac{2}{9}(16-5x-13x^2)$
Hence, a corresponding orthonormal set of vector is:
$\{1-2x+2x^2,\frac{2}{9}(16-5x-13x^2)\}$
