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The given function is:

\[
f(x) = x^4 + 3x^3 + 7x^2 + 15x + 10
\]

The problem asks to evaluate \( f(x) \) at two specific values of \( x \), namely \( x = -2 \) and \( x = -1 \).

### 1. Evaluating \( f(x) \) at \( x = -2 \):
\[
f(-2) = (-2)^4 + 3(-2)^3 + 7(-2)^2 + 15(-2) + 10
\]
Breaking it down:
\[
= 16 + 3(-8) + 7(4) + 15(-2) + 10
\]
\[
= 16 - 24 + 28 - 30 + 10
\]
\[
= 0
\]

Thus, \( f(-2) = 0 \).

### 2. Evaluating \( f(x) \) at \( x = -1 \):
\[
f(-1) = (-1)^4 + 3(-1)^3 + 7(-1)^2 + 15(-1) + 10
\]
Breaking it down:
\[
= 1 + 3(-1) + 7(1) + 15(-1) + 10
\]
\[
= 1 - 3 + 7 - 15 + 10
\]
\[
= 0
\]

Thus, \( f(-1) = 0 \).

### Conclusion:
- \( f(-2) = 0 \)
- \( f(-1) = 0 \)

Both \( x = -2 \) and \( x = -1 \) are roots of the given polynomial.

Would you like more details or have any questions about this? Here are some follow-up questions that can expand on the topic:

1. How do you factor a polynomial once you know its roots?
2. What methods can be used to find the roots of a fourth-degree polynomial?
3. How does the Rational Root Theorem help in finding possible roots?
4. Can all fourth-degree polynomials be factored easily?
5. How can you graph the function \( f(x) = x^4 + 3x^3 + 7x^2 + 15x + 10 \)?

**Tip:** When evaluating polynomials at specific points, always simplify each step incrementally to avoid mistakes with negative signs and powers.

Evaluate the function f(x) = x^4 + 3x^3 + 7x^2 + 15x + 10 at x = -2 and x = -1.

Learn how to evaluate the polynomial function f(x) = x^4 + 3x^3 + 7x^2 + 15x + 10 at x = -2 and x = -1, with step-by-step explanations. Understand the process of polynomial evaluation and find the roots of this fourth-degree function.

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