Let $P$ be the parabola with equation $y = x^2$ and let $Q = (20, 14)$ There are real numbers $r$ and $s$ such that the line through $Q$ with slope $m$ does not intersect $P$ if and only if $r < m < s$. What is $r + s?$

$ \textbf{(A)} 1 \qquad \textbf{(B)} 26 \qquad \textbf{(C)} 40 \qquad \textbf{(D)} 52 \qquad \textbf{(E)} 80 \qquad $
1. Let the line through \( Q = (20, 14) \) be denoted by \(\ell\). The equation of the line \(\ell\) can be written in the slope-intercept form as \( y = mx + b \).

2. Since the line passes through the point \( Q = (20, 14) \), we can substitute these coordinates into the line equation to find \( b \):
   \[
   14 = 20m + b \implies b = 14 - 20m
   \]

3. We want to find the conditions under which the line \(\ell\) does not intersect the parabola \( P \) given by \( y = x^2 \). The intersection points of the line and the parabola are given by solving the equation:
   \[
   mx + b = x^2 \implies x^2 - mx - b = 0
   \]

4. For the line to not intersect the parabola, the quadratic equation \( x^2 - mx - b = 0 \) must have no real solutions. This occurs when the discriminant of the quadratic equation is less than zero. The discriminant \(\Delta\) of the quadratic equation \( ax^2 + bx + c = 0 \) is given by:
   \[
   \Delta = b^2 - 4ac
   \]
   For our equation \( x^2 - mx - b = 0 \), \( a = 1 \), \( b = -m \), and \( c = -b \). Thus, the discriminant is:
   \[
   \Delta = (-m)^2 - 4(1)(-b) = m^2 + 4b
   \]

5. We substitute \( b = 14 - 20m \) into the discriminant:
   \[
   \Delta = m^2 + 4(14 - 20m) = m^2 + 56 - 80m
   \]

6. For the line to not intersect the parabola, we need:
   \[
   m^2 + 56 - 80m < 0
   \]

7. We solve the inequality \( m^2 - 80m + 56 < 0 \) by finding the roots of the corresponding quadratic equation:
   \[
   m^2 - 80m + 56 = 0
   \]

8. Using the quadratic formula \( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1 \), \( b = -80 \), and \( c = 56 \):
   \[
   m = \frac{80 \pm \sqrt{80^2 - 4 \cdot 1 \cdot 56}}{2 \cdot 1} = \frac{80 \pm \sqrt{6400 - 224}}{2} = \frac{80 \pm \sqrt{6176}}{2}
   \]

9. Simplifying further:
   \[
   m = \frac{80 \pm 2\sqrt{1544}}{2} = 40 \pm \sqrt{1544}
   \]

10. Let \( r = 40 - \sqrt{1544} \) and \( s = 40 + \sqrt{1544} \). The line does not intersect the parabola if and only if \( r < m < s \).

11. The sum of \( r \) and \( s \) is:
   \[
   r + s = (40 - \sqrt{1544}) + (40 + \sqrt{1544}) = 80
   \]

Conclusion:
\[
r + s = \boxed{80}
\]