Answer :
EXPLANATION.
α and β are the zeroes of the polynomial.
f(x) = x² - p(x + 1) - c.
As we know that,
Concepts :
α and β are the zeroes of the quadratic polynomial.
Sum of the zeroes of the quadratic polynomial.
⇒ α + β = -b/a.
Products of the zeroes of the quadratic polynomial.
⇒ αβ = c/a.
Using this concepts in this question, we get.
⇒ f(x) = x² - p(x + 1) - c.
We can write expression as,
⇒ f(x) = x² - px - p - c.
⇒ f(x) = x² - (p)x - (p + c).
Sum of the zeroes of the quadratic polynomial.
⇒ α + β = p. - - - - - (1).
Products of the zeroes of the quadratic polynomial.
⇒ αβ = - (p + c). - - - - - (2).
To find : (α + 1)(β + 1).
⇒ (α + 1)(β + 1) = α(β + 1) + 1(β + 1).
⇒ (α + 1)(β + 1) = αβ + α + β + 1.
⇒ (α + 1)(β + 1) = (αβ) + (α + β) + 1.
Put the value in the expression, we get.
⇒ (α + 1)(β + 1) = - (p + c) + p + 1.
⇒ (α + 1)(β + 1) = - p - c + p + 1.
⇒ (α + 1)(β + 1) = - c + 1.
⇒ (α + 1)(β + 1) = 1 - c.
∴ The value of (α + 1)(β + 1) is equal to (1 - c).
Option [B] is correct answer.