If α, β are the roots of x² - p(x+1) - c = 0, then find the value of (α + 1)(β + 1) — JEE Mathematics

Rewrite the given equation in standard form:

$$x^2 - px - (p + c) = 0$$

From the coefficients, we can find the sum and product of the roots:

$$\alpha + \beta = p$$

$$\alpha\beta = -(p + c)$$

Now expand the expression we want to evaluate:

$$(\alpha + 1)(\beta + 1) = \alpha\beta + \alpha + \beta + 1$$

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

$$(\alpha + 1)(\beta + 1) = -(p + c) + p + 1 = -p - c + p + 1 = 1 - c$$

Answer: $1 - c$