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Hint: As it is given that (x+2) is a factor of ${{x}^{2}}+ax+2b=0$ , we can say that -2 is one of the roots of the given equation. So, put the value of x to be -2 and get an equation in terms of a and b. Now solve the equation you get and the equation a+b=4 to get the values of a and b.

Complete step-by-step answer:

Let us start the solution to the above question. It is given that (x+2) is a factor of ${{x}^{2}}+ax+2b=0$ , which implies that -2 is one of the roots of the given quadratic equation. As -2 is the root of the equation it must satisfy the quadratic equation.

$\therefore {{x}^{2}}+ax+2b=0$

$\Rightarrow {{\left( -2 \right)}^{2}}-2a+2b=0$

$\Rightarrow 4-2a+2b=0$

$\Rightarrow a-b=2.......(i)$

Also, it is given in the question that:

$a+b=4...........(ii)$

If we add both, equation (i) and equation (ii), we get

$a+b+a-b=2+4$

$\Rightarrow 2a=6$

$\Rightarrow a=3$

Now if we will put the value of a in equation (ii), we will get

$a+b=4$

$\Rightarrow 3+b=4$

$\Rightarrow b=1$

Therefore, we can conclude that the value of a is 3 and b is equal to 1.

Note: If you want, you can solve the above question using the relation between the roots and the coefficients of a quadratic equation, but that would be lengthy. But if you are asked about the other root of the equation as well then the method of relation of root with the coefficient of the quadratic equation would be our first priority.

Complete step-by-step answer:

Let us start the solution to the above question. It is given that (x+2) is a factor of ${{x}^{2}}+ax+2b=0$ , which implies that -2 is one of the roots of the given quadratic equation. As -2 is the root of the equation it must satisfy the quadratic equation.

$\therefore {{x}^{2}}+ax+2b=0$

$\Rightarrow {{\left( -2 \right)}^{2}}-2a+2b=0$

$\Rightarrow 4-2a+2b=0$

$\Rightarrow a-b=2.......(i)$

Also, it is given in the question that:

$a+b=4...........(ii)$

If we add both, equation (i) and equation (ii), we get

$a+b+a-b=2+4$

$\Rightarrow 2a=6$

$\Rightarrow a=3$

Now if we will put the value of a in equation (ii), we will get

$a+b=4$

$\Rightarrow 3+b=4$

$\Rightarrow b=1$

Therefore, we can conclude that the value of a is 3 and b is equal to 1.

Note: If you want, you can solve the above question using the relation between the roots and the coefficients of a quadratic equation, but that would be lengthy. But if you are asked about the other root of the equation as well then the method of relation of root with the coefficient of the quadratic equation would be our first priority.

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