The value of $b - c$ for which the identity $f(x + 1) - f(x) = 8x + 3$ is satisfied where $f(x) = b{x^2} + cx + d$ is

Hint: First we will put $x = x + 1$ in the given function equation $f(x) = b{x^2} + cx + d$, then will subtract $f(x) = b{x^2} + cx + d$ from the resultant equation. After solving, the simplified form will compare with $f(x + 1) - f(x) = 8x + 3$ to get the value of $b$ and $c$. To get the required answer will subtract $c$ from $b$.

Formula Used: ${(a + b)^2} = {a^2} + 2ab + {b^2}$

Complete step by step solution: Given, $f(x) = b{x^2} + cx + d$-----(1)
Put $x = x + 1$ in the equation (1)
$f(x + 1) = b{(x + 1)^2} + c(x + 1) + d$
Using formula ${(a + b)^2} = {a^2} + 2ab + {b^2}$
$f(x + 1) = b({x^2} + 2x + 1) + cx + c + d$
After solving, we get
$f(x + 1) = b{x^2} + 2bx + b + cx + c + d$-----(2)
Subtracting equation (1) from equation (2)
$f(x + 1) - f(x) = b{x^2} + 2bx + b + cx + c + d - b{x^2} - cx - d$
After simplifying, we will get
$f(x + 1) - f(x) = 2bx + b + c$-----(3)
Given, $f(x + 1) - f(x) = 8x + 3$-----(4)
On comparing equation (3) and equation (4)
$2bx + b + c = 8x + 3$
$2bx = 8x$
Dividing both sides by $2x$
$b = 4$
$b + c = 3$
Putting the value of b
$4 + c = 3$
Subtracting 4 from both the sides
$c = - 1$
So, $b - c = 4 - ( - 1)$
$b - c = 5$

Hence, the value of $b - c$ is $5$

Note: Students should first understand the question carefully before solving that. And should put the correct value of $x$ in the functional equation and do calculations carefully to get the exact answer.