Given the equations:

$7\alpha^2 - 5\alpha = -49$ $7\beta^2 - 5\beta = -49$

we can rearrange each equation to:

$7\alpha^2 - 5\alpha + 49 = 0$ $7\beta^2 - 5\beta + 49 = 0$

These are quadratic equations in $\alpha$ and $\beta$. Let's solve the quadratic equation for $\alpha$:

$7\alpha^2 - 5\alpha + 49 = 0$

Using the quadratic formula $\alpha = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$:

For our equation, $a = 7$, $b = -5$, and $c = 49$.

$\alpha = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 7 \cdot 49}}{2 \cdot 7}$ $\alpha = \frac{5 \pm \sqrt{25 - 1372}}{14}$ $\alpha = \frac{5 \pm \sqrt{-1347}}{14}$

Since the discriminant $(-1347)$ is negative, the solutions for $\alpha$ are complex numbers. Let:

$\alpha = \frac{5 \pm i\sqrt{1347}}{14}$

Similarly, for $\beta$:

$\beta = \frac{5 \pm i\sqrt{1347}}{14}$

Since $\alpha$ and $\beta$ have the same form, let's denote:

$\alpha = \frac{5 + i\sqrt{1347}}{14}$ $\beta = \frac{5 - i\sqrt{1347}}{14}$

Now, we need to find the value of $\alpha^2\beta + \alpha\beta^2$. First, calculate $\alpha\beta$:

$\alpha\beta = \left( \frac{5 + i\sqrt{1347}}{14} \right) \left( \frac{5 - i\sqrt{1347}}{14} \right)$ $\alpha\beta = \frac{(5 + i\sqrt{1347})(5 - i\sqrt{1347})}{196}$ $\alpha\beta = \frac{25 - (i\sqrt{1347})^2}{196}$ $\alpha\beta = \frac{25 + 1347}{196}$ $\alpha\beta = \frac{1372}{196}$ $\alpha\beta = 7$

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

$\alpha^2\beta + \alpha\beta^2 = \alpha\beta(\alpha + \beta)$

We know $\alpha\beta = 7$, so we need to find $\alpha + \beta$:

$\alpha + \beta = \frac{5 + i\sqrt{1347}}{14} + \frac{5 - i\sqrt{1347}}{14}$ $\alpha + \beta = \frac{10}{14}$ $\alpha + \beta = \frac{5}{7}$

Thus,

$\alpha^2\beta + \alpha\beta^2 = 7 \left( \frac{5}{7} \right)$ $\alpha^2\beta + \alpha\beta^2 = 5$

So, the value of $\alpha^2\beta + \alpha\beta^2$ is $5$.

Would you like further details or have any questions? Here are 5 numbered questions for you to ask next:

1. Can you solve another quadratic equation?
2. What are the properties of complex numbers?
3. How do you derive the quadratic formula?
4. Can you explain the significance of the discriminant in quadratic equations?
5. How do you solve systems of equations involving quadratic terms?

Tip: Always check the discriminant first when solving quadratic equations to determine the nature of the roots.