If `a^(2)-4a+1=4`, then the value of `(a^(3)-a^(2)+a-1)/(a^(2)-1)(a^(2)ne1)`

Explanation

To solve the problem step by step, we start with the equation given in the question and proceed to find the required value.

### Step 1: Solve the given equation
We start with the equation:
\[ a^2 - 4a + 1 = 4 \]

Subtract 4 from both sides:
\[ a^2 - 4a + 1 - 4 = 0 \]
\[ a^2 - 4a - 3 = 0 \]

### Step 2: Factor the quadratic equation
Next, we need to factor the quadratic equation:
\[ a^2 - 4a - 3 = 0 \]

To factor, we look for two numbers that multiply to -3 (the constant term) and add to -4 (the coefficient of \(a\)). The numbers -5 and 1 satisfy this condition:
\[ (a - 5)(a + 1) = 0 \]

### Step 3: Find the roots
Now, we can find the roots of the equation:
Setting each factor to zero gives us:
1. \( a - 5 = 0 \) → \( a = 5 \)
2. \( a + 1 = 0 \) → \( a = -1 \)

### Step 4: Substitute the roots into the expression
We need to evaluate the expression:
\[ \frac{a^3 - a^2 + a - 1}{(a^2 - 1)} \]

First, we simplify the denominator:
\[ a^2 - 1 = (a - 1)(a + 1) \]

### Step 5: Evaluate for \( a = 5 \)
Substituting \( a = 5 \):
1. Calculate the numerator:
\[ a^3 - a^2 + a - 1 = 5^3 - 5^2 + 5 - 1 \]
\[ = 125 - 25 + 5 - 1 = 104 \]

2. Calculate the denominator:
\[ a^2 - 1 = 5^2 - 1 = 25 - 1 = 24 \]

Thus, the expression becomes:
\[ \frac{104}{24} = \frac{13}{3} \]

### Step 6: Evaluate for \( a = -1 \)
Now substituting \( a = -1 \):
1. Calculate the numerator:
\[ a^3 - a^2 + a - 1 = (-1)^3 - (-1)^2 + (-1) - 1 \]
\[ = -1 - 1 - 1 - 1 = -4 \]

2. Calculate the denominator:
\[ a^2 - 1 = (-1)^2 - 1 = 1 - 1 = 0 \]
(This means the expression is undefined for \( a = -1 \))

### Final Result
The only valid value we found is for \( a = 5 \):
\[ \frac{13}{3} \]