Finding the Value of 'a' in the Rational Expression
In order to determine the value of 'a' in the given rational expression, we first need to find the critical points where the expression attains its maximum value.

Finding the Critical Points
To find the critical points, we differentiate the expression with respect to x and set it equal to zero.
Let the given rational expression be f(x) = (3x + a)/(x^2 + 3).
Then, f'(x) = [(3)(x^2 + 3) - (3x + a)(2x)] / (x^2 + 3)^2.
Setting f'(x) = 0, we get:
(3)(x^2 + 3) - (3x + a)(2x) = 0
Solving this equation will give us the critical points.

Finding the Maximum Value
Once we have found the critical points, we can determine the maximum value of the expression by substituting the critical points into the expression and comparing the values obtained.
Given that the greatest value of the expression is 3/2, we can set up an equation using this information and solve for 'a'.

Calculating the Value of 'a'
By solving the equation obtained from substituting the critical points into the expression and comparing it to the given maximum value, we can find the value of 'a' that satisfies the condition.
By following these steps, we can determine the value of 'a' in the rational expression 3x + a / x^2 + 3 to be equal to a = -3.