Step 1: Consider the given function
I = \frac{x^2 - 16}{\sqrt{x^2 + 3x - 2}}
Step 2: Factorize the numerator and simplify the denominator.
x^2 - 16 = (x - 4)(x + 4)
x^2 + 3x - 2 can be factored into:
\sqrt{(x + 4)(x - 1)}
Step 3: Put the factored expressions back into the original fraction:
I = \frac{(x - 4)(x + 4)}{\sqrt{(x + 4)(x - 1)}}
Step 4: Simplify the expression:
I = \sqrt{\frac{(x - 4)(x + 4)^2}{(x - 1)(x + 4)}}
I = \frac{(x + 4)(x - 4)}{\sqrt{(x + 4)(x - 1)}}
As a result, the simplified solution is:
I = \frac{x^2 - 16}{\sqrt{x^2 + 3x - 2}}
Therefore, the final answer is:
I = \frac{x^2 - 16}{\sqrt{x^2 + 3x - 2}}