$$\mathop {\lim }\limits_{x \to \infty } \left( {{{{x^2} + x + 1} \over {x + 1}} - ax - b} \right) = 4$$

$$ \Rightarrow \mathop {\lim }\limits_{x \to \infty } {{{x^2} + x + 1 - a{x^2} - ax - bx - b} \over {x + 1}} = 4$$

$$ \Rightarrow \mathop {\lim }\limits_{x \to \infty } {{{x^2}(1 - a) + x(1 - a - b) + (1 - b)} \over {x + 1}} = 4$$

Here, we make degree of N^r = degree of D^r

$$\therefore$$ $$1 - a = 0$$

and $$\mathop {\lim }\limits_{x \to \infty } {{x(1 - a - b) + (1 - b)} \over {x + 1}} = 4$$

$$ \Rightarrow 1 - a - b = 4$$

$$ \Rightarrow b = - 4$$