Let $(2, 3)$ be the largest open interval in which the function $f(x) = 2 \log_e (x - 2) - x^2 + a x + 1$ is strictly increasing and $(b, c)$ be the largest open interval in which the function $g(x) = (x - 1)^3 (x + 2 - a)^2$ is strictly decreasing. Then $100(a + b - c)$ is equal to:

Step 1: Find the derivative of f(x) and determine 'a'

To find where the function $f(x)$ is strictly increasing, we need to find its first derivative, $f'(x)$, and set it greater than zero. The derivative of $\log_e(x-2)$ is $\frac{1}{x-2}$, and the derivative of $-x^2 + ax + 1$ is $-2x + a$. Since $f(x)$ is strictly increasing in $(2, 3)$, $f'(x) > 0$ for $x \in (2, 3)$. The critical points occur when $f'(x) = 0$. Given that $(2, 3)$ is the largest interval where $f(x)$ is strictly increasing, $x=3$ must be a critical point, meaning $f'(3) = 0$.

Step 2: Solve for 'a'

Substitute $x = 3$ into the derivative $f'(x)$ and set it equal to zero to solve for the constant $a$. This gives us the value of $a$ that makes $x=3$ a critical point.

Step 3: Find the derivative of g(x)

Next, we need to find the derivative of $g(x)$ to determine where it is strictly decreasing. We use the product rule for differentiation. Substitute $a=4$ into the expression for $g(x)$ before or after differentiation.

Step 4: Simplify g'(x) and find critical points

Factor out common terms from $g'(x)$ to simplify it. Then substitute $a=4$ into the simplified expression. The critical points are where $g'(x) = 0$.

Step 5: Determine the intervals for g(x) decreasing

Substitute $a=4$ into the factored form of $g'(x)$ and simplify the expression. The critical points are $x=1$, $x=2$, and $x=\frac{8}{5} = 1.6$. We need to analyze the sign of $g'(x)$ in the intervals defined by these critical points. Since $(x-1)^2 \ge 0$, the sign of $g'(x)$ depends on $(x-2)(5x-8)$. $g(x)$ is strictly decreasing when $g'(x) < 0$. This occurs when $(x-2)$ and $(5x-8)$ have opposite signs, which is for $x \in (\frac{8}{5}, 2)$. Thus, $b = \frac{8}{5}$ and $c = 2$.

Step 6: Calculate the final value

Finally, substitute the values of $a$, $b$, and $c$ into the expression $100(a + b - c)$ and calculate the result.

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