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The problem asks to determine the real zeros, their multiplicities, and to sketch the graph of the given polynomial function:

\[
f(x) = 2x^4 + x^3 - 6x^2
\]

Here are the steps to solve it:

### 1. **Find the real zeros of the polynomial**:
Factor the polynomial to find its zeros.

\[
f(x) = 2x^4 + x^3 - 6x^2 = x^2(2x^2 + x - 6)
\]

Now, factor the quadratic part \( 2x^2 + x - 6 \):

\[
2x^2 + x - 6 = (2x - 3)(x + 2)
\]

Thus, the factored form of \( f(x) \) is:

\[
f(x) = x^2(2x - 3)(x + 2)
\]

### 2. **Determine the real zeros**:
From the factored form \( f(x) = x^2(2x - 3)(x + 2) \), the real zeros are:

- \( x = 0 \) (with multiplicity 2, since \( x^2 \) gives a double root),
- \( x = \frac{3}{2} \) (with multiplicity 1),
- \( x = -2 \) (with multiplicity 1).

### 3. **Check the graphs provided**:
We now need to match this to the correct graph by looking for:
- A double root at \( x = 0 \) (which means the graph will touch the x-axis at \( x = 0 \) but not cross it),
- Single roots at \( x = \frac{3}{2} \) and \( x = -2 \) (the graph will cross the x-axis at these points).

The graph that matches these criteria is **Option A**. It shows a touch at \( x = 0 \) (indicating multiplicity 2) and crossings at \( x = \frac{3}{2} \) and \( x = -2 \).

### Answer:
- **Real Zeros**: \( 0 \) (multiplicity 2), \( \frac{3}{2} \) (multiplicity 1), \( -2 \) (multiplicity 1).
- **Correct Graph**: **Option A**.

Would you like further details or explanations on any part of this solution?

Here are 5 related questions you might want to explore:
1. How does the multiplicity of a zero affect the shape of the graph at that point?
2. What are the steps for factoring higher-degree polynomials?
3. Can a polynomial of degree 4 have more than 4 real zeros?
4. How can you determine end behavior of a polynomial from its leading term?
5. How does the sign of the leading coefficient affect the graph's direction?

**Tip**: When a polynomial has a zero with an even multiplicity, the graph "bounces" off the x-axis at that zero instead of crossing it.

Determine the real zeros and state the multiplicity of any repeated zeros, then sketch the graph for the polynomial f(x) = 2x^4 + x^3 - 6x^2.

Learn how to determine the real zeros and their multiplicity for the polynomial f(x) = 2x^4 + x^3 - 6x^2. This guide explains polynomial factoring, finding zeros, and understanding root multiplicity, along with graph interpretation. Suitable for students in Grades 10-12.

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