Question algebraically. Answer Attempt 1 out of 2 (1) Add a zero $$ \odot $$ No zeros $$ x=x^{3}+7 x^{2}+2 x-40 $$ and $$ x+4 $$ is a factor of $$ f(x) $$, then find all of the zeros of $$ f(x) $$

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To find the zeros of the polynomial $$ f(x) = x^3 + 7x^2 + 2x - 40 $$ given that $$ x + 4 $$ is a factor, we can use polynomial long division or synthetic division to divide $$ f(x) $$ by $$ x + 4 $$.

### Step 1: Perform Synthetic Division

We will use synthetic division with the root $$ x = -4 $$:

1. Write down the coefficients of $$ f(x) $$: $$ 1, 7, 2, -40 $$.
2. Set up synthetic division:

```
-4 |  1   7   2  -40
   |      -4  -12   40
   ---------------------
     1   3  -10    0
```

### Step 2: Write the Result

The result of the synthetic division is $$ 1x^2 + 3x - 10 $$ with a remainder of $$ 0 $$. Thus, we can write:

$$
f(x) = (x + 4)(x^2 + 3x - 10)
$$

### Step 3: Factor the Quadratic

Next, we need to factor the quadratic $$ x^2 + 3x - 10 $$. We can do this by finding two numbers that multiply to $$-10$$ and add to $$3$$. The numbers $$5$$ and $$-2$$ work:

$$
x^2 + 3x - 10 = (x + 5)(x - 2)
$$

### Step 4: Write the Complete Factorization

Now we can write the complete factorization of $$ f(x) $$:

$$
f(x) = (x + 4)(x + 5)(x - 2)
$$

### Step 5: Find the Zeros

To find the zeros of $$ f(x) $$, we set each factor equal to zero:

1. $$ x + 4 = 0 $$  →  $$ x = -4 $$
2. $$ x + 5 = 0 $$  →  $$ x = -5 $$
3. $$ x - 2 = 0 $$  →  $$ x = 2 $$

### Conclusion

The zeros of the polynomial $$ f(x) = x^3 + 7x^2 + 2x - 40 $$ are:

$$
\boxed{-4, -5, 2}
$$
The zeros of $$ f(x) $$ are -4, -5, and 2.

Question algebraically. Answer Attempt 1 out of 2 (1) Add a zero \( \odot \) No zeros \( x=x^{3}+7 x^{2}+2 x-40 \) and \( x+4 \) is a factor of \( f(x) \), then find all of the zeros of \( f(x) \)

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