What is the answer to this Maths q.?

Be careful with your notation. As written the division takes precedence, so your expression would equivalent to:
(26/4) + √3
= 13/2 + √3
= 6.5 + √3

However, that's obviously not what you meant because it doesn't simplify to a form with integers values of a and b.

What you intended to write was:
26/(4 + √3)

Do you see why the parentheses are necessary?

Okay, from there, just multiply top and bottom by the conjugate of 4 + √3 --> namely 4 - √3
26(4 - √3)
--------------------
(4 + √3)(4 - √3)

26(4 - √3)
---------------
16 - (√3)²

26(4 - √3)
---------------
16 - 3

26(4 - √3)
---------------
..... 13

2(4 - √3)

Answer:
8 - 2√3

a = 8, b = -2

26/4+√3 = 13/2+√3 = a+b√3 with a=13/2 and b=1.
Unfortunately, a is not an integer.

Cannot be expressed with integer coefficients as presented. However if parentheses are added such that the term is:

26/(4+√3)

then the denominator can be rationalized by multiplying by the conjugate of the denominator:

26(4-√3)/[(4+√3)(4-√3)]
= (104 - 26√3)/(16-3)
= 104/13 - 26√3/13
= 8 - 2√3

26 ........4 - √3
--------- x ---------
4 + √3 ... 4 - √3

104 - 26√3
---------------
13

8 -2√3

a=8
b=-2

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26/(4 + √3)
= 26(4 - √3)/(4 + √3)(4 - √3)
= 2(4 - √3)

It's already in that form.
If you meant 26/(4 + √3), then
26/(4 + √3) * (4 - √3)/(4 - √3) =
(104 - 26√3)/(16 - 3) =
(104/7 - 26√3)/13 =
104/13 - 26√3/13 =
8 - 2√3

Could it be that should actually be shown as :-
26 / [ 4 + √3 ] ?

If so , this becomes :-

26 [ 4 - √ 3 ]
----------------------
[ 4 + √3 ] [ 4 - √ 3 ]

26 [ 4 - √ 3 ]
----------------------
16 - 9

26 [ 4 - √ 3 ]
----------------------
7

(26/7) [ 4 - √ 3 ]