Match each entry in List-I to the correct entry in List-II and...

For (P):
Since $\alpha$ and $\beta$ are roots of $x^2 + x + 1 = 0$, we have $\alpha^2 + \alpha + 1 = 0 \Rightarrow \alpha + 1 = -\alpha^2$ and $\alpha^3 = 1$.
$\dfrac{1}{(\alpha+1)^{2026}} = \dfrac{1}{(-\alpha^2)^{2026}} = \dfrac{1}{\alpha^{4052}}$
Since $4052 = 3 \times 1350 + 2$, $\alpha^{4052} = (\alpha^3)^{1350} \cdot \alpha^2 = \alpha^2$.
Thus, $\dfrac{1}{\alpha^2} = \dfrac{\alpha^3}{\alpha^2} = \alpha$.
Similarly, $\dfrac{1}{(\beta+1)^{2026}} = \beta$.
The quadratic equation with roots $\alpha$ and $\beta$ is the original equation $x^2 + x + 1 = 0$.
So, (P) $\rightarrow$ (1).

For (Q):
$\dfrac{1}{(\alpha+1)^{2027}} = \dfrac{1}{(-\alpha^2)^{2027}} = -\dfrac{1}{\alpha^{4054}}$
Since $4054 = 3 \times 1351 + 1$, $\alpha^{4054} = \alpha$.
Thus, $-\dfrac{1}{\alpha} = -\beta$.
Similarly, the other root is $-\alpha$.
Sum of roots $= -\alpha - \beta = -(\alpha + \beta) = -(-1) = 1$.
Product of roots $= (-\alpha)(-\beta) = \alpha\beta = 1$.
The quadratic equation is $x^2 - x + 1 = 0$.
So, (Q) $\rightarrow$ (2).

For (R):
Since $\gamma$ and $\delta$ are roots of $x^2 - x + 1 = 0$, we have $\gamma^2 - \gamma + 1 = 0 \Rightarrow \gamma - 1 = \gamma^2$ and $\gamma^3 = -1$.
$\dfrac{1}{(\gamma-1)^{2026}} = \dfrac{1}{(\gamma^2)^{2026}} = \dfrac{1}{\gamma^{4052}}$
$\gamma^{4052} = (\gamma^3)^{1350} \cdot \gamma^2 = (-1)^{1350} \cdot \gamma^2 = \gamma^2$.
Thus, $\dfrac{1}{\gamma^2} = \dfrac{\gamma}{\gamma^3} = -\gamma$.
Similarly, the second term is $-\delta$.
The sum is $-\gamma - \delta = -(\gamma + \delta) = -(1) = -1$.
So, (R) $\rightarrow$ (4).

For (S):
Since $p$ and $r$ are roots of $x^2 + x - 1 = 0$, we have $x^2 + x = 1 \Rightarrow x(x+1) = 1 \Rightarrow x+1 = \dfrac{1}{x}$.
$\dfrac{1}{(p+1)^3} + \dfrac{1}{(r+1)^3} = p^3 + r^3$
We know $p+r = -1$ and $pr = -1$.
$p^3 + r^3 = (p+r)^3 - 3pr(p+r) = (-1)^3 - 3(-1)(-1) = -1 - 3 = -4$.
So, (S) $\rightarrow$ (5).

Answer: (P) $\rightarrow$ (1), (Q) $\rightarrow$ (2), (R) $\rightarrow$ (4), (S) $\rightarrow$ (5)