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The problem asks to find the polynomial \( g(x) \) given the relationship:

\[
g(x) = -2f(x) - 8
\]

where \( f(x) = 7x^3 - 8x^2 + 12 \).

### Step-by-step solution:

1. First, substitute the expression for \( f(x) \) into the equation for \( g(x) \):

\[
g(x) = -2(7x^3 - 8x^2 + 12) - 8
\]

2. Now, distribute the \(-2\) across the terms inside the parentheses:

\[
g(x) = -2(7x^3) + (-2)(-8x^2) + (-2)(12) - 8
\]
\[
g(x) = -14x^3 + 16x^2 - 24 - 8
\]

3. Combine the constants \(-24\) and \(-8\):

\[
g(x) = -14x^3 + 16x^2 - 32
\]

Thus, the polynomial \( g(x) \) in standard form is:

\[
g(x) = -14x^3 + 16x^2 - 32
\]

Would you like more details on any of the steps? Here are 5 related questions you might find useful:

1. How do you perform polynomial multiplication and distribution?
2. What does it mean for a polynomial to be in "standard form"?
3. How do constants combine when simplifying polynomial expressions?
4. Can this method be extended to higher-order polynomials?
5. What role do negative signs play in polynomial operations?

**Tip**: Always distribute multiplication carefully, especially with negative signs, to avoid sign errors in the solution.

The function f(x) is given by f(x) = 7x^3 − 8x^2 + 12. Write the equation for polynomial g(x) in standard form such that g(x) = −2f(x) − 8.

Learn how to transform the polynomial function f(x) = 7x^3 − 8x^2 + 12 to g(x) = −2f(x) − 8 with a step-by-step solution. This problem demonstrates polynomial operations and algebraic transformation techniques suitable for Grades 9-12.

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