Given the quadratic equation $ax^2 + bx + c = 0$, where $\alpha$ and $\beta$ are the roots, we are required to find the value of the following expression:

$\frac{\alpha}{a\beta + b} + \frac{\beta}{a\alpha + b}$

Step 1: Use Vieta's formulas

From Vieta's formulas, for a quadratic equation $ax^2 + bx + c = 0$, the sum and product of the roots are given by:

$\alpha + \beta = -\frac{b}{a}$ $\alpha \beta = \frac{c}{a}$

Step 2: Simplify the given expression

We start by simplifying the expression:

$\frac{\alpha}{a\beta + b} + \frac{\beta}{a\alpha + b}$

To simplify this, we need a common denominator:

$\text{Common denominator} = (a\beta + b)(a\alpha + b)$

Thus, the expression becomes:

$\frac{\alpha(a\alpha + b) + \beta(a\beta + b)}{(a\beta + b)(a\alpha + b)}$

Expanding the numerator:

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

We can group terms:

$a(\alpha^2 + \beta^2) + b(\alpha + \beta)$

Step 3: Simplify $\alpha^2 + \beta^2$

The expression $\alpha^2 + \beta^2$ can be written in terms of $\alpha + \beta$ and $\alpha \beta$:

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

Substitute the values from Vieta's formulas:

$\alpha^2 + \beta^2 = \left(-\frac{b}{a}\right)^2 - 2 \times \frac{c}{a} = \frac{b^2}{a^2} - \frac{2c}{a}$

So the numerator becomes:

$a\left(\frac{b^2}{a^2} - \frac{2c}{a}\right) + b\left(-\frac{b}{a}\right) = \frac{ab^2}{a^2} - 2ac + \frac{-b^2}{a}$

Simplifying:

$\frac{b^2}{a} - 2ac - \frac{b^2}{a} = -2ac$

Step 4: Denominator Simplification

The denominator is:

$(a\beta + b)(a\alpha + b) = a^2\alpha \beta + ab(\alpha + \beta) + b^2$

Substitute $\alpha \beta = \frac{c}{a}$ and $\alpha + \beta = -\frac{b}{a}$:

$a^2\frac{c}{a} + ab\left(-\frac{b}{a}\right) + b^2 = ac - b^2 + b^2 = ac$

Step 5: Final expression

Finally, the expression simplifies to:

$\frac{-2ac}{ac} = -2$

So, the value of the expression is $\boxed{-2}$.

Would you like more details on any step, or have any questions?

Here are some related questions:

1. How does Vieta's formula relate to the roots of quadratic equations?
2. What are the conditions for the discriminant of a quadratic equation?
3. How do you derive the formula for $\alpha^2 + \beta^2$ using $\alpha + \beta$ and $\alpha \beta$?
4. Can you generalize the result for different forms of quadratic equations?
5. What is the geometric interpretation of the roots of a quadratic equation?

Tip: Always check if you can simplify expressions involving roots by using Vieta's formulas, which relate the coefficients of the polynomial to the sum and product of its roots.