Factorise : `x^(4)+5x^(3)+5x^(2)-5x-6`

Options

A. `(x^(2)-1)(x^(2)+6)`
B. `(x-1)(x+2)^(3)`
C. `(x^(2)-1)(x+3)(x+2)`
D. `(x-1)(x+2)(x^(2)+3)`

Answer

To factorize the polynomial \( x^4 + 5x^3 + 5x^2 - 5x - 6 \), we will follow a systematic approach.

### Step-by-Step Solution:

1. **Identify the Polynomial**:
We are given the polynomial \( P(x) = x^4 + 5x^3 + 5x^2 - 5x - 6 \).

2. **Look for Rational Roots**:
We can use the Rational Root Theorem to test for possible rational roots. The possible rational roots could be the factors of the constant term (-6) divided by the factors of the leading coefficient (1). The possible rational roots are \( \pm 1, \pm 2, \pm 3, \pm 6 \).

3. **Test Possible Roots**:
We will test these values to see if any of them make \( P(x) = 0 \).

- Testing \( x = 1 \):
\[ P(1) = 1^4 + 5(1^3) + 5(1^2) - 5(1) - 6 = 1 + 5 + 5 - 5 - 6 = 0 \]
Thus, \( x = 1 \) is a root.

4. **Factor Out \( (x - 1) \)**:
Since \( x = 1 \) is a root, we can factor the polynomial by dividing \( P(x) \) by \( (x - 1) \) using synthetic division or polynomial long division.

Performing synthetic division:
\[ \begin{array}{r|rrrrr} 1 & 1 & 5 & 5 & -5 & -6 \\ & & 1 & 6 & 11 & 6 \\ \hline & 1 & 6 & 11 & 6 & 0 \\ \end{array} \]
The result of the division is \( x^3 + 6x^2 + 11x + 6 \).

5. **Factor the Resulting Polynomial**:
Now we need to factor \( x^3 + 6x^2 + 11x + 6 \). We can again look for rational roots among \( \pm 1, \pm 2, \pm 3, \pm 6 \).

- Testing \( x = -2 \):
\[ P(-2) = (-2)^3 + 6(-2)^2 + 11(-2) + 6 = -8 + 24 - 22 + 6 = 0 \]
Thus, \( x = -2 \) is also a root.

6. **Factor Out \( (x + 2) \)**:
We can factor \( x^3 + 6x^2 + 11x + 6 \) by dividing it by \( (x + 2) \).

Performing synthetic division:
\[ \begin{array}{r|rrrr} -2 & 1 & 6 & 11 & 6 \\ & & -2 & -8 & -6 \\ \hline & 1 & 4 & 3 & 0 \\ \end{array} \]
The result is \( x^2 + 4x + 3 \).

7. **Factor the Quadratic**:
Now we need to factor \( x^2 + 4x + 3 \). This can be factored as:
\[ x^2 + 4x + 3 = (x + 1)(x + 3) \]

8. **Combine All Factors**:
Thus, the complete factorization of the original polynomial is:
\[ P(x) = (x - 1)(x + 2)(x + 1)(x + 3) \]

### Final Answer:
The factorization of \( x^4 + 5x^3 + 5x^2 - 5x - 6 \) is:
\[ (x - 1)(x + 2)(x + 1)(x + 3) \]