For all `c ne pm (1)/(5), (5c^(2)-24c-5)/(1-25c^(2))=`

Options

A. `(5c+1)/(5+c)`
B. `(5+c)/(5c+1)`
C. `-(5c-1)/(5-c)`
D. `(5-c)/(5c-1)`

Explanation

To solve the equation \(\frac{5c^2 - 24c - 5}{1 - 25c^2}\), we will follow these steps:

### Step 1: Factor the numerator
The numerator is \(5c^2 - 24c - 5\). We need to factor this quadratic expression.

To factor \(5c^2 - 24c - 5\), we look for two numbers that multiply to \(5 \times (-5) = -25\) and add up to \(-24\). The numbers \(-25\) and \(1\) work.

So we can rewrite the expression:
\[ 5c^2 - 25c + c - 5 \]
Now, we can group the terms:
\[ (5c^2 - 25c) + (c - 5) \]
Factoring out the common terms:
\[ 5c(c - 5) + 1(c - 5) \]
Now, we can factor out \((c - 5)\):
\[ (5c + 1)(c - 5) \]

### Step 2: Factor the denominator
The denominator is \(1 - 25c^2\), which is a difference of squares:
\[ 1 - (5c)^2 = (1 - 5c)(1 + 5c) \]

### Step 3: Rewrite the expression
Now we can rewrite the original expression using the factored forms:
\[ \frac{(5c + 1)(c - 5)}{(1 - 5c)(1 + 5c)} \]

### Step 4: Simplify the expression
Notice that \(1 - 5c\) can be rewritten as \(-(5c - 1)\):
\[ \frac{(5c + 1)(c - 5)}{-(5c - 1)(1 + 5c)} \]
This allows us to cancel out the common factors. However, we need to be careful about the signs.

Multiplying the numerator and denominator by \(-1\) gives:
\[ \frac{-(5c + 1)(c - 5)}{(5c - 1)(1 + 5c)} \]

### Step 5: Final expression
This simplifies to:
\[ \frac{-(5c + 1)(c - 5)}{(5c - 1)(1 + 5c)} \]

### Step 6: Conclusion
Thus, the simplified expression is:
\[ \frac{5 - c}{5c - 1} \]