This problem asks us to find the zeros of the quadratic function:

$f(x) = 3x^2 + 5x + 2$

Step-by-step Solution:

1. Find the Zeros of the Function

The zeros of a function occur where $f(x) = 0$. Setting the equation to zero:

$3x^2 + 5x + 2 = 0$

We solve this quadratic equation using factoring, as it is factorable.

2. Factor the Quadratic Expression

The quadratic expression $3x^2 + 5x + 2$ can be factored as:

$(3x + 2)(x + 1) = 0$

3. Solve for $x$

Using the zero-product property ($ab = 0 \implies a = 0 \text{ or } b = 0$):

• $3x + 2 = 0$ $3x = -2 \quad \Rightarrow \quad x = -\frac{2}{3}$

• $x + 1 = 0$ $x = -1$

4. Conclusion

The zeros of the function are: $x = -\frac{2}{3} \quad \text{and} \quad x = -1$

Answer:

Move the numbers $-\frac{2}{3}$ and $-1$ to the answer blanks.

Would you like further details on any step or concept?

Here are 5 related questions to consider:

1. How do you factor quadratics that are not easily factorable?
2. What is the quadratic formula, and when should it be used?
3. How can you verify the zeros of a quadratic function?
4. What does the graph of $f(x) = 3x^2 + 5x + 2$ look like, and where do the zeros appear on it?
5. How does the discriminant help in determining the type of solutions for a quadratic equation?

Tip: Always check your factoring by expanding the factors back to the original quadratic!