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Let's solve the problem step by step.

### Given:
- \( f(x) = 4x^2 + 3x \)
- \( a = -1 \)

#### Part a: Find \( f'(a) \)

To find \( f'(a) \), we need to first differentiate \( f(x) \) and then substitute \( x = a = -1 \) into the derivative.

1. Differentiate \( f(x) \):
\[
f'(x) = \frac{d}{dx}(4x^2) + \frac{d}{dx}(3x)
\]
\[
f'(x) = 8x + 3
\]

2. Substitute \( a = -1 \) into \( f'(x) \):
\[
f'(-1) = 8(-1) + 3 = -8 + 3 = -5
\]

Thus, \( f'(a) = -5 \).

#### Part b: Determine the equation of the tangent line to the graph at \( a = -1 \)

The equation of the tangent line at \( x = a \) is given by:
\[
y - f(a) = f'(a)(x - a)
\]

1. First, find \( f(-1) \):
\[
f(-1) = 4(-1)^2 + 3(-1) = 4(1) - 3 = 1
\]
Thus, \( f(-1) = 1 \).

2. The slope \( f'(a) = -5 \) from part a.

3. The equation of the tangent line:
\[
y - 1 = -5(x + 1)
\]
\[
y - 1 = -5x - 5
\]
\[
y = -5x - 4
\]

Thus, the equation of the tangent line is \( y = -5x - 4 \).

Let me know if you'd like further clarification! Here are some related questions:

1. What are the general steps to differentiate a polynomial function?
2. How does the derivative relate to the slope of the tangent line?
3. Can you explain the geometric interpretation of a tangent line?
4. How do we compute derivatives for more complex functions like trigonometric or exponential functions?
5. What role does the second derivative play in graphing the curvature of a function?

**Tip:** Always check whether you need to compute a derivative implicitly when variables are dependent on each other!

For the function f(x) = 4x^2 + 3x, a = -1, find f'(a) and determine the equation of the tangent line at a.

Learn how to differentiate the function f(x) = 4x^2 + 3x and find the equation of the tangent line at a = -1. This problem covers key calculus concepts like derivatives and tangent lines. Suitable for Grades 10-12.

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