On dividing $x^{3} - 3x^{2} + x + 2$ by a polynomial $g(x)$, the quotient and remainder were $x - 2$ and $-2x + 4$ respectively. Find $g(x)$.
$x^{2}-x+1$
Dividend $f(x)=x^{3}-3x^{2}+x+2$, quotient $q(x)=x-2$, remainder $r(x)=-2x+4$. Use $f(x)=g(x)q(x)+r(x)$: \[ g(x)=\frac{f(x)-r(x)}{q(x)} =\frac{x^{3}-3x^{2}+x+2-(-2x+4)}{x-2} =\frac{x^{3}-3x^{2}+3x-2}{x-2}. \] Divide: $(x-2)(x^{2}-x+1)=x^{3}-3x^{2}+3x-2$. Thus $g(x)=\boxed{x^{2}-x+1}$.
If the equation \( a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \) has equal roots, where \( a + c = 15 \) and \( b = \frac{36}{5} \), then \( a^2 + c^2 \) is equal to .
Find the missing code:
L1#1O2~2, J2#2Q3~3, _______, F4#4U5~5, D5#5W6~6