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

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:

$2b = 8 \implies b = 4$

$b + c = 3 \implies 4 + c = 3 \implies c = -1$

The values $b = 4$, $c = -1$ correspond to option B.

The solution is $\boxed{\text{B}}$