Question

If a. and B are zeroes of a given polynomial
f(x) = x2-p (x + 2) - c, then find the value of
(a + 2) (B + 2)

Answer 1

Rewriting the question...

"If a and b are zeroes of a given polynomial f(x) = x² - p(x + 2) - c, then find the value of (a + 2)(b + 2)."

So,

f(x) = x² - p(x + 2) - c

=> f(x) = x² - px - 2p - c

=> f(x) = x² - px - (2p + c)

Let A = 1 ; B = - p ; C = - (2p + c)

Since a and b are zeroes of f(x), we get,

a + b = - B/A = - (- p)/1 = p

And

ab = C/A = - (2p + c)/1 = - (2p + c)

Now,

(a + 2)(b + 2)

=> ab + 2a + 2b + 4

=> ab + 2(a + b) + 4

=> - (2p + c) + 2p + 4

=> - 2p - c + 2p + 4

=> 4 - c

Hence,

$\Large\boxed{\mathsf{(a+2)(b+2)=4-c}}$

Answer 2

f(x)=x^2-px-2p-c

a+b=p

ab=-(2p+c)

(a+2)(b+2)=ab+2(a+b)+4

-2p-c+2p+4

4-c

hope it helps