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The image contains a math problem involving functions. Here’s the breakdown of what is visible on the board:

Given functions:
1. \( f(x) = -x^2 + 3x + 6 \)
2. \( g(x) = 3x^2 - 5x \)
3. \( h(x) = 6x + 1 \)

Tasks:
a. Calculate \( f(x) + g(x) \).
b. Calculate \( (f - g)(x) \).
c. Calculate \( (f \times g)(x) \).

Let's go through each task:

### a. \( f(x) + g(x) \):
We add the two functions:
\[
f(x) + g(x) = (-x^2 + 3x + 6) + (3x^2 - 5x)
\]
Simplify:
\[
= -x^2 + 3x^2 + 3x - 5x + 6
= 2x^2 - 2x + 6
\]

### b. \( (f - g)(x) \):
We subtract \( g(x) \) from \( f(x) \):
\[
(f - g)(x) = (-x^2 + 3x + 6) - (3x^2 - 5x)
\]
Simplify:
\[
= -x^2 - 3x^2 + 3x + 5x + 6
= -4x^2 + 8x + 6
\]

### c. \( (f \times g)(x) \):
We multiply \( f(x) \) and \( g(x) \):
\[
(f \times g)(x) = (-x^2 + 3x + 6)(3x^2 - 5x)
\]
Using distribution (FOIL):
\[
= (-x^2)(3x^2) + (-x^2)(-5x) + (3x)(3x^2) + (3x)(-5x) + (6)(3x^2) + (6)(-5x)
\]
Simplifying each term:
\[
= -3x^4 + 5x^3 + 9x^3 - 15x^2 + 18x^2 - 30x
\]
Combine like terms:
\[
= -3x^4 + 14x^3 + 3x^2 - 30x
\]

Would you like further explanation or any clarifications?

### Related Questions:
1. How would you solve for \( f(x) + h(x) \)?
2. Can you evaluate \( (g \times h)(x) \)?
3. What is the result of dividing \( f(x) \) by \( g(x) \)?
4. How would you find the roots of \( f(x) + g(x) \)?
5. Can you find the derivative of \( f(x) \)?

**Tip**: Always combine like terms carefully in polynomials to avoid mistakes during simplification.

The image contains a math problem involving functions. The tasks include performing operations such as addition, subtraction, and multiplication on the given functions f(x) = -x^2 + 3x + 6 and g(x) = 3x^2 - 5x.

Learn how to perform addition, subtraction, and multiplication of polynomial functions. The functions f(x) = -x^2 + 3x + 6 and g(x) = 3x^2 - 5x are explored with step-by-step solutions.

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