Let's solve each of the requested problems:

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#12: Zeros: $-2$ and $1 + 2i$

To write the polynomial with real coefficients, remember that complex roots come in conjugate pairs. Therefore, the zeros are $-2$, $1 + 2i$, and $1 - 2i$.

The factors corresponding to these zeros are: $(x + 2), \quad (x - (1 + 2i)), \quad (x - (1 - 2i)).$

Combine the complex conjugate factors: $(x - (1 + 2i))(x - (1 - 2i)) = (x - 1 - 2i)(x - 1 + 2i) = (x - 1)^2 - (2i)^2 = (x - 1)^2 + 4.$

Now multiply this with $(x + 2)$: $(x + 2)((x - 1)^2 + 4) = (x + 2)(x^2 - 2x + 1 + 4) = (x + 2)(x^2 - 2x + 5).$

Expand: $x(x^2 - 2x + 5) + 2(x^2 - 2x + 5) = x^3 - 2x^2 + 5x + 2x^2 - 4x + 10 = x^3 + x^2 + x + 10.$

Final polynomial: $f(x) = x^3 + x^2 + x + 10.$

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#14: Zeros: $-1$ (multiplicity 3), $3$ (multiplicity 1)

The factors for the zeros are: $(x + 1)^3 \quad \text{and} \quad (x - 3).$

Multiply these together: $(x + 1)^3 = (x + 1)(x + 1)(x + 1) = (x^2 + 2x + 1)(x + 1) = x^3 + 3x^2 + 3x + 1.$

Now multiply by $(x - 3)$: $(x^3 + 3x^2 + 3x + 1)(x - 3).$

Expand: $x(x^3 + 3x^2 + 3x + 1) - 3(x^3 + 3x^2 + 3x + 1) = x^4 + 3x^3 + 3x^2 + x - 3x^3 - 9x^2 - 9x - 3.$

Combine like terms: $f(x) = x^4 + 0x^3 - 6x^2 - 8x - 3.$

Final polynomial: $f(x) = x^4 - 6x^2 - 8x - 3.$

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#16: Zeros: $2$ (multiplicity 2), $-2 + i$ (multiplicity 1)

Since $-2 + i$ is a root, its conjugate $-2 - i$ must also be a root.

The factors corresponding to these zeros are: $(x - 2)^2, \quad (x - (-2 + i)), \quad (x - (-2 - i)).$

Simplify the complex conjugate pair: $(x - (-2 + i))(x - (-2 - i)) = (x + 2 - i)(x + 2 + i) = (x + 2)^2 - (i)^2 = (x + 2)^2 + 1 = x^2 + 4x + 5.$

Now multiply this with $(x - 2)^2$: $(x - 2)^2(x^2 + 4x + 5).$

First expand $(x - 2)^2$: $(x - 2)^2 = (x - 2)(x - 2) = x^2 - 4x + 4.$

Now multiply this with $(x^2 + 4x + 5)$: $(x^2 - 4x + 4)(x^2 + 4x + 5).$

Expand: $x^2(x^2 + 4x + 5) - 4x(x^2 + 4x + 5) + 4(x^2 + 4x + 5).$

$= x^4 + 4x^3 + 5x^2 - 4x^3 - 16x^2 - 20x + 4x^2 + 16x + 20.$

Combine like terms: $f(x) = x^4 - 7x^2 - 4x + 20.$

Final polynomial: $f(x) = x^4 - 7x^2 - 4x + 20.$

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Recap of Results:

1. #12: $f(x) = x^3 + x^2 + x + 10$
2. #14: $f(x) = x^4 - 6x^2 - 8x - 3$
3. #16: $f(x) = x^4 - 7x^2 - 4x + 20$

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Would you like more detailed steps or additional questions?
Here are five related problems for practice:

1. How do you verify that a polynomial has the given roots?
2. Write a polynomial where the zeros are $0, 3$, and $-3$.
3. Solve for a polynomial with $1 + 3i$ and $1 - 3i$ as roots.
4. Write a polynomial with $-1$ (multiplicity 2) and $2$ (multiplicity 1).
5. What happens to the degree of a polynomial when you increase a root's multiplicity?

Tip: Always check for conjugate pairs when working with complex roots and real coefficients!