The equations $x^2-5x+6=0$ and $x^2+\mathrm{ax}+b=0$, have exactly one common root, and the other root of the second equation is –4. Among the following, which are the possible values of a and b?

To find the possible values of $a$ and $b$, we can solve the problem step-by-step.

Step 1: Find the roots of the first quadratic equation.
The first equation is given by:
$x^2-5x+6=0$

We can factor this quadratic equation by splitting the middle term:
$x^2-2x-3x+6=0$
$x(x-2)-3(x-2)=0$
$(x-2)(x-3)=0$

Thus, the roots of the first equation are:
$x=2$ and $x=3$

Step 2: Analyze the second quadratic equation.
The second equation is:
$x^2+ax+b=0$

We are given that the two equations share exactly one common root, and the other root of this second equation is -4. This leads to two possible cases depending on which root is shared.

Case 1: The common root is $x=2$.
In this case, the roots of the second equation are 2 and -4.
Using the relationship between the roots and coefficients of a quadratic equation:
The sum of the roots is:
$2+(-4)=-a$
$-2=-a$
$a=2$

The product of the roots is:
$2\times (-4)=b$
$b=-8$

So, one possible pair of values is $a=2$ and $b=-8$.

Case 2: The common root is $x=3$.
In this case, the roots of the second equation are 3 and -4.
The sum of the roots is:
$3+(-4)=-a$
$-1=-a$
$a=1$

The product of the roots is:
$3\times (-4)=b$
$b=-12$

So, another possible pair of values is $a=1$ and $b=-12$.

Conclusion:
Comparing our findings with the given options, the pair $a=2,b=-8$ corresponds directly to the correct option.