The values of b and c for which the identity f (x+1)- f (x)=8 x+3 is satisfied, where f (x)= b x^2+ c x+ d , are

Step-by-step solution

Given f(x) = bx^2 + cx + d , determine b and c such that f(x+1) - f(x) = 8x + 3 . Compute f(x+1) by substitution: f(x+1) = b(x+1)^2 + c(x+1) + d = b(x^2 + 2x + 1) + cx + c + d = bx^2 + (2b + c)x + (b + c + d) The difference simplifies to: f(x+1) - f(x) = [bx^2 + (2b + c)x + (b + c + d)] - [bx^2 + cx + d] = 2bx + (b + c) Set equal to the given expression: 2bx + (b + c) = 8x + 3 Equating coefficients yields: 2b = 8 b = 4 b + c = 3 4 + c = 3 c = -1 The values b = 4 , c = -1 correspond to option B . The solution is B