To find the values of a and b in the given expression, we need to simplify the expression first. Let's break down the given expression step by step:

Step 1: Simplify the denominator
The denominator of the expression is (4 - 3√5). To simplify it, we can multiply the numerator and denominator by the conjugate of the denominator, which is (4 + 3√5).

(4 - 3√5) * (4 + 3√5) = 16 - 9 * 5 = 16 - 45 = -29

So, the simplified denominator is -29.

Step 2: Simplify the numerator
The numerator of the expression is 4 * 3√5, which can be written as 12√5.

Step 3: Simplify the entire expression
Now, we can rewrite the given expression as:

12√5 / -29

Step 4: Express the expression in the form of a + b√5
To express the expression in the form of a + b√5, we need to rationalize the denominator.

Multiply both the numerator and denominator by -1 to make the denominator positive:

(12√5 / -29) * (-1 / -1) = -12√5 / 29

Now, we can express the expression as:

-12 / 29 * √5

So, the value of a is -12/29 and the value of b is 1.

Explanation:

- The expression is simplified by rationalizing the denominator.
- The simplified expression is then expressed in the form of a + b√5.
- The values of a and b are determined as -12/29 and 1, respectively.
- The final expression is -12 / 29 * √5.