Question 661685
There might be a shorter way to do this, but here's how I solved it.  
Given that {{{x^2 - x + 1 = 0}}} has roots at x=p and x=q, we know that 
{{{x^2 - x + 1 = 0}}} can be factored into {{{(x - p)(x - q) = 0}}}, so equivalently
{{{x^2 - x + 1 = (x - p)(x - q) = x^2 -(p + q)*x + (p * q) = 0}}}. 
This gives us {{{p + q = 1}}} and {{{p * q = 1}}}
Similarly, {{{3*x^2 - 2*x + 3 = 0}}} can be factored into {{{3(x - r)(x - s) = 0}}}.  
We can divide both sides of each by 3 to get
{{{x^2 - (2/3)*x + 1 = (x - r)(x - s) = x^2 - (r + s)*x + (r * s) = 0}}}.
This gives us {{{r + s = 2/3}}} and {{{r * s = 1}}}
The next thing we need is:
{{{(p + q)^2 = p^2 + 2pq + q2 = (p^2 + q^2) + 2*p*q}}}
{{{(p + q)^2 - 2*p*q = p^2 + q^2}}}
And we know from above that {{{p+q = 1}}} and {{{p*q = 1}}} so
{{{1^2 - 2*1 = p^2 + q^2}}} or
{{{p^2 + q^2 = 1 - 2 = -1}}}
Similarly, we can get
{{{r^2 + s^2 = (r + s)^2 - 2*r*s = (2/3)^2 - 2*1 = 4/9 - 2 = -14/9}}}
Finally, we need one last step:
{{{(p + q)(r + s) = pr + ps + qr + qs}}}
But since {{{p+q = 1}}} and {{{r+s = 2/3}}} we have
{{{pr + ps + qr + qs = 1*(2/3) = 2/3}}}
Now we can multiply out (p-r)(q-r)(p-s)(q-s)
Let's regroup:
{{{(p-r)*(q-r)*(p-s)*(q-s) = ((p-r)*(q-s))*((q-r)*(p-s)) }}}
{{{(p-r)*(q-r)*(p-s)*(q-s) = (pq - ps - qr + rs)*(pq - qs - pr + rs) }}}
Substituting pq=1 and rs=1:
{{{(p-r)*(q-r)*(p-s)*(q-s) = (1 - ps - qr + 1)(1 - qs - pr + 1)}}}
{{{(p-r)*(q-r)*(p-s)*(q-s) = (2 - ps - qr)(2 - qs - pr)}}}
{{{(p-r)*(q-r)*(p-s)*(q-s) = 4 - 2*qs - 2*pr - 2*ps + pq*s^2 + p^2*rs - 2*qr + q^2*rs + pq*r^2}}}
Again, substituting pq=1 and rs=1:
{{{(p-r)*(q-r)*(p-s)*(q-s) = 4 - 2*qs - 2*pr - 2*ps + s^2 + p^2 - 2qr + q^2 + r^2}}}
{{{(p-r)*(q-r)*(p-s)*(q-s) = 4 - 2(pr + ps + qr + qs) + (p^2 + q^2) + (r^2 + s^2)}}}
From above, {{{pr + ps + qr + qs = 2/3}}}, {{{p^2 + q^2 = -1}}}, and {{{r^2 + s^2 = -14/9}}}
{{{(p-r)*(q-r)*(p-s)*(q-s) = 4 - 2*(2/3) + (-1) + (-14/9)}}}
{{{(p-r)*(q-r)*(p-s)*(q-s) = 4 - 4/3 - 1 -14/9}}}
{{{(p-r)*(q-r)*(p-s)*(q-s) = 36/9 - 12/9 - 9/9 -14/9}}}
{{{(p-r)*(q-r)*(p-s)*(q-s) = (36 - 12 - 9 - 14)/9}}}
{{{(p-r)*(q-r)*(p-s)*(q-s) = 1/9}}}