Given the quadratic equation 3x2 − 11x + 4 = 0, with roots alpha and beta. Find the equation whose roots are X1 = alpha2 + (1)/(beta) and X2 = (1)/(alpha) + beta2.

$$S = \left(\alpha^2 + \frac{1}{\beta}\right) + \left(\frac{1}{\alpha} + \beta^2\right)$$ $$S = \alpha^2 + \beta^2 + \frac{1}{\alpha} + \frac{1}{\beta}$$

We know that $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta$.
And $\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta}$.

First, calculate $\alpha^2 + \beta^2$:

$$\alpha^2 + \beta^2 = \left(\frac{11}{3}\right)^2 - 2\left(\frac{4}{3}\right)$$ $$\alpha^2 + \beta^2 = \frac{121}{9} - \frac{8}{3}$$ $$\alpha^2 + \beta^2 = \frac{121}{9} - \frac{24}{9} = \frac{97}{9}$$

Next, calculate $\frac{1}{\alpha} + \frac{1}{\beta}$:

$$\frac{1}{\alpha} + \frac{1}{\beta} = \frac{11/3}{4/3} = \frac{11}{4}$$

Now, substitute these values back into the expression for $S$:

$$S = \frac{97}{9} + \frac{11}{4}$$

To add these fractions, find a common denominator, which is 36.

$$S = \frac{97 \times 4}{9 \times 4} + \frac{11 \times 9}{4 \times 9}$$ $$S = \frac{388}{36} + \frac{99}{36} = \frac{487}{36}$$

$$P = \left(\alpha^2 + \frac{1}{\beta}\right) \left(\frac{1}{\alpha} + \beta^2\right)$$

Expand the product:

$$P = \alpha^2 \cdot \frac{1}{\alpha} + \alpha^2 \beta^2 + \frac{1}{\beta} \cdot \frac{1}{\alpha} + \frac{1}{\beta} \beta^2$$ $$P = \alpha + (\alpha \beta)^2 + \frac{1}{\alpha \beta} + \beta$$

Rearrange the terms:

$$P = (\alpha + \beta) + (\alpha \beta)^2 + \frac{1}{\alpha \beta}$$

Substitute the values of $\alpha + \beta$ and $\alpha \beta$:

$$P = \frac{11}{3} + \left(\frac{4}{3}\right)^2 + \frac{1}{4/3}$$ $$P = \frac{11}{3} + \frac{16}{9} + \frac{3}{4}$$

To add these fractions, find a common denominator, which is 36.

$$P = \frac{11 \times 12}{3 \times 12} + \frac{16 \times 4}{9 \times 4} + \frac{3 \times 9}{4 \times 9}$$ $$P = \frac{132}{36} + \frac{64}{36} + \frac{27}{36}$$ $$P = \frac{132 + 64 + 27}{36} = \frac{223}{36}$$

Substitute the calculated values of $S$ and $P$:

$$x^2 - \frac{487}{36}x + \frac{223}{36} = 0$$

Multiply the entire equation by 36 to eliminate the denominators:

$$36x^2 - 487x + 223 = 0$$