$ ext{The expression } x^3 - 3x^2 - 9x - 5 ext{ has factors :}$

Question

$	ext{The expression } x^3 - 3x^2 - 9x - 5 	ext{ has factors :}$

Accepted Answer

To determine why Option 2 is the correct factorization of the expression $x^3 - 3x^2 - 9x - 5$, we need to verify that this expression can be factored into $(x+1)^2(x-5)$.

### Verification of Option 2:

1. **Expand $(x+1)^2(x-5)$:**

- First, expand $(x+1)^2$:
     $$
     (x+1)^2 = x^2 + 2x + 1
     $$

- Next, multiply this result by $(x-5)$:
     $$
     (x^2 + 2x + 1)(x-5) = x^2(x-5) + 2x(x-5) + 1(x-5)
     $$

- Calculate each term:
     $$
     x^2(x-5) = x^3 - 5x^2
     $$
     $$
     2x(x-5) = 2x^2 - 10x
     $$
     $$
     1(x-5) = x - 5
     $$

- Combine all terms:
     $$
     x^3 - 5x^2 + 2x^2 - 10x + x - 5 = x^3 - 3x^2 - 9x - 5
     $$

The expansion matches the original expression, confirming that $(x+1)^2(x-5)$ is indeed a factorization of $x^3 - 3x^2 - 9x - 5$.

### Explanation of Incorrect Options:

1. **Option 1: $(x-1)^2(x+5)$**
   - Expanding $(x-1)^2(x+5)$ results in a different polynomial:
     $$
     (x-1)^2 = x^2 - 2x + 1
     $$
     $$
     (x^2 - 2x + 1)(x+5) = x^3 + 5x^2 - 2x^2 - 10x + x + 5 = x^3 + 3x^2 - 9x + 5
     $$
   - This does not match the original expression $x^3 - 3x^2 - 9x - 5$.

2. **Option 3: $(x-2)^2(x+1)$**
   - Expanding $(x-2)^2(x+1)$ results in:
     $$
     (x-2)^2 = x^2 - 4x + 4
     $$
     $$
     (x^2 - 4x + 4)(x+1) = x^3 + x^2 - 4x^2 - 4x + 4x + 4 = x^3 - 3x^2 + 4
     $$
   - This does not match the original expression.

3. **Option 4: $(x+1)(x+2)(x+16)$**
   - Expanding $(x+1)(x+2)(x+16)$ results in:
     $$
     (x+1)(x+2) = x^2 + 3x + 2
     $$
     $$
     (x^2 + 3x + 2)(x+16) = x^3 + 16x^2 + 3x^2 + 48x + 2x + 32 = x^3 + 19x^2 + 50x + 32
     $$
   - This does not match the original expression.

In conclusion, Option 2 is the correct factorization of the expression $x^3 - 3x^2 - 9x - 5$ because it accurately expands to the original polynomial, while the other options do not.