$\displaystyle \frac{6x^{2} + ax + a + 2}{x^{2} + x - 6} = \frac{6x^2 + ax + a+2}{(x+3)(x-2)}$

We can see that the (x+3) factor is causing the problem if we directly evaluated the limit by plugging in x = -3. So, let's

**assume** that the numerator is divisible by (x+3) so that

$\displaystyle \frac{(x+3)(\text{stuff})}{(x+3)(x+2)}$

the (x+3)'s will cancel and all that remains is to plug in x = -3 directly.

So, we do long division and you should get the remainder: $\displaystyle 56 - 2a$

Since we want (x+3) to factor in nicely, we can assume that the remainder is 0. i.e. $\displaystyle 56 - 2a = 0 \quad \Rightarrow a = 28$

So we have: $\displaystyle \frac{6x^{2} + 28x + 30}{(x+3)(x+2)} = \frac{2(x+3)(3x+5)}{(x+3)(x+2)}$

You know where to go from there

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Or if you learned l'hopital's, you could go that route and set the numerator equal to 0 and solve for a. Then differentiate and solve the limit.