Given: alpha − beta = 8 and alpha² + beta² = 74.

Here's a step-by-step solution to each problem:

: Quadratic Equation Roots
Given: $\alpha - \beta = 8$ and $\alpha^2 + \beta^2 = 74$.
We need to find a quadratic equation with integer coefficients whose roots are $r_1 = 2\alpha + \beta$ and $r_2 = 2\beta + \alpha$.

Step 1: Find $\alpha\beta$.
We know that $(\alpha - \beta)^2 = \alpha^2 - 2\alpha\beta + \beta^2$.
Substitute the given values:
$8^2 = 74 - 2\alpha\beta$
$64 = 74 -