https://file.aiquickdraw.com/mathbot/file/1726238342333t4pc83r3.jpg

Let's review the steps to determine if the solution is correct:

### The problem asks to find \( a \) such that \( f'(1) = -3 \).

#### Given expression for \( f'(x) \):
\[
f'(x) = 3ax^2 + 10x + 2
\]

#### Step 1: Substituting \( x = 1 \) into \( f'(x) \):
\[
f'(1) = 3a \cdot 1^2 + 10 \cdot 1 + 2
\]
This simplifies to:
\[
f'(1) = 3a + 10 + 2
\]
\[
f'(1) = 3a + 12
\]

#### Step 2: Solving for \( a \) with \( f'(1) = -3 \):
\[
3a + 12 = -3
\]
Subtracting 12 from both sides:
\[
3a = -3 - 12
\]
\[
3a = -15
\]
Dividing by 3:
\[
a = \frac{-15}{3}
\]
\[
a = -5
\]

### Conclusion:
The solution presented is **correct**. The steps and the value \( a = -5 \) are accurate.

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Would you like further clarification on any step? Here are 5 follow-up questions:

1. Why does substituting \( x = 1 \) in the derivative equation help solve for \( a \)?
2. How does the power rule apply to calculating the derivative \( f'(x) \)?
3. Can you explain the algebra involved in solving for \( a \) step-by-step?
4. What is the significance of the constant \( f'(1) = -3 \) in this context?
5. What would the equation \( f'(x) \) look like if \( a = 2 \)?

**Tip**: Always double-check calculations involving basic algebra to ensure no small mistakes slip in!

Learn how to solve for the variable a in a calculus problem where f'(1) = -3. This step-by-step solution explains how to calculate derivatives, apply algebraic operations, and find the correct value for a. Suitable for students in Grades 10-12.

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