If
x+2
2x
​
+
2x+4
1
​
−
2x
2
+4x
12−x
​
=a+
x
b
​
, where a and b are integers, then the value of ab=_____.

Let's simplify the expression on the Left-Hand Side (LHS):
LHS=
x+2
2x
​
+
2(x+2)
1
​
−
2x(x+2)
12−x
​

The common denominator for all three terms is 2x(x+2). Converting each fraction to have this denominator:

LHS=
2x(x+2)
2x⋅2x+1⋅x−(12−x)
​

LHS=
2x(x+2)
4x
2
+x−12+x
​

LHS=
2x(x+2)
4x
2
+2x−12
​

Factorize the numerator 4x
2
+2x−12:

4x
2
+2x−12=2(2x
2
+x−6)=2(2x
2
+4x−3x−6)=2[2x(x+2)−3(x+2)]=2(2x−3)(x+2)
Substitute this back into the LHS expression:

LHS=
2x(x+2)
2(2x−3)(x+2)
​

Canceling the common terms 2 and (x+2) from the numerator and denominator:

LHS=
x
2x−3
​
=
x
2x
​
−
x
3
​
=2−
x
3
​

Comparing this to the Right-Hand Side (RHS), which is a+
x
b
​
:

a=2andb=−3
Therefore, the value of ab is:

ab=2×(−3)=−6