We are given the functions:

• $f(x) = 4x^2 + 5x$
• $g(x) = 28 - 4x^2$

We need to perform several operations on these functions and determine their domains.

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1. Find and simplify $(f + g)(x)$

$(f + g)(x) = f(x) + g(x)$

Substitute the given functions:

$(f + g)(x) = (4x^2 + 5x) + (28 - 4x^2)$

Simplify the expression:

$(f + g)(x) = 4x^2 - 4x^2 + 5x + 28 = 5x + 28$

Domain of $(f + g)(x)$:

Since $5x + 28$ is a linear function, there are no restrictions on $x$. The domain is all real numbers:

$\text{Domain of } (f + g)(x) = (-\infty, \infty)$

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2. Find and simplify $(f - g)(x)$

$(f - g)(x) = f(x) - g(x)$

Substitute the given functions:

$(f - g)(x) = (4x^2 + 5x) - (28 - 4x^2)$

Simplify the expression:

$(f - g)(x) = 4x^2 + 5x - 28 + 4x^2 = 8x^2 + 5x - 28$

Domain of $(f - g)(x)$:

Since $8x^2 + 5x - 28$ is a quadratic function, there are no restrictions on $x$. The domain is all real numbers:

$\text{Domain of } (f - g)(x) = (-\infty, \infty)$

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3. Find and simplify $(fg)(x)$

$(fg)(x) = f(x) \cdot g(x)$

Substitute the given functions:

$(fg)(x) = (4x^2 + 5x)(28 - 4x^2)$

Now expand the product:

$(fg)(x) = 4x^2(28 - 4x^2) + 5x(28 - 4x^2)$

$= 4x^2(28) - 4x^2(4x^2) + 5x(28) - 5x(4x^2)$

$= 112x^2 - 16x^4 + 140x - 20x^3$

Rearrange terms in descending powers of $x$:

$(fg)(x) = -16x^4 - 20x^3 + 112x^2 + 140x$

Domain of $(fg)(x)$:

Since $(fg)(x)$ is a polynomial, there are no restrictions on $x$. The domain is all real numbers:

$\text{Domain of } (fg)(x) = (-\infty, \infty)$

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Summary:

1. $(f + g)(x) = 5x + 28$, Domain: $(-\infty, \infty)$
2. $(f - g)(x) = 8x^2 + 5x - 28$, Domain: $(-\infty, \infty)$
3. $(fg)(x) = -16x^4 - 20x^3 + 112x^2 + 140x$, Domain: $(-\infty, \infty)$

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Would you like further clarification on any part of this process? Here are some related questions for further exploration:

1. What are the roots of $f(x) + g(x) = 0$?
2. How do you factor $f(x) - g(x)$?
3. Can you determine the critical points of $(fg)(x)$?
4. What is the behavior of $(fg)(x)$ as $x \to \infty$?
5. What is the degree of $(fg)(x)$?

Tip: Always check the domain for each function before performing any operations, especially when dealing with rational or logarithmic functions!