Factorise and simplify : `(2x^(5)-2x)/(2x^(3)+2x)` .The answer is

Options

A. `x^(2)-1`
B. `x^(2)+1`
C. `x-1`
D. `x+1`

Answer

To factorise and simplify the expression \(\frac{2x^5 - 2x}{2x^3 + 2x}\), we will follow these steps:

### Step 1: Factor out common terms
First, we will factor out the common terms from both the numerator and the denominator.

**Numerator:**
\[ 2x^5 - 2x = 2x(x^4 - 1) \]

**Denominator:**
\[ 2x^3 + 2x = 2x(x^2 + 1) \]

### Step 2: Rewrite the expression
Now we can rewrite the original expression using the factored forms:
\[ \frac{2x(x^4 - 1)}{2x(x^2 + 1)} \]

### Step 3: Cancel common factors
Next, we can cancel the common factor \(2x\) from the numerator and the denominator:
\[ \frac{x^4 - 1}{x^2 + 1} \]

### Step 4: Factor \(x^4 - 1\)
Now, we will factor \(x^4 - 1\) using the difference of squares:
\[ x^4 - 1 = (x^2 - 1)(x^2 + 1) \]

### Step 5: Substitute back into the expression
Now we can substitute this back into our expression:
\[ \frac{(x^2 - 1)(x^2 + 1)}{x^2 + 1} \]

### Step 6: Cancel the common factor again
We can cancel \(x^2 + 1\) from the numerator and the denominator:
\[ x^2 - 1 \]

### Step 7: Final factorization
Finally, we can factor \(x^2 - 1\) again using the difference of squares:
\[ x^2 - 1 = (x - 1)(x + 1) \]

### Final Answer
Thus, the simplified and factorised form of the expression is:
\[ (x - 1)(x + 1) \]

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