Let's solve each of the given expressions step by step.

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

We are given:

• $g(x) = x + 7$
• $h(x) = 3x^2 + 23x + 14$

To find $g(x) + h(x)$, simply add these two functions:

$g(x) + h(x) = (x + 7) + (3x^2 + 23x + 14)$

Combine like terms:

$g(x) + h(x) = 3x^2 + (23x + x) + (14 + 7)$ $g(x) + h(x) = 3x^2 + 24x + 21$

So, the simplified expression is:

$g(x) + h(x) = 3x^2 + 24x + 21$

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

We are given:

• $g(x) = x + 7$
• $h(x) = 3x^2 + 23x + 14$

To find $h(x) - g(x)$, subtract the two functions:

$h(x) - g(x) = (3x^2 + 23x + 14) - (x + 7)$

Distribute the negative sign and combine like terms:

$h(x) - g(x) = 3x^2 + 23x + 14 - x - 7$ $h(x) - g(x) = 3x^2 + (23x - x) + (14 - 7)$ $h(x) - g(x) = 3x^2 + 22x + 7$

So, the simplified expression is:

$h(x) - g(x) = 3x^2 + 22x + 7$

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3. Find and simplify $f(x) \times h(x)$

We are given:

• $f(x) = 8x$
• $h(x) = 3x^2 + 23x + 14$

To find $f(x) \times h(x)$, multiply these two functions:

$f(x) \times h(x) = 8x(3x^2 + 23x + 14)$

Distribute $8x$ to each term in $h(x)$:

$f(x) \times h(x) = 8x \times 3x^2 + 8x \times 23x + 8x \times 14$ $f(x) \times h(x) = 24x^3 + 184x^2 + 112x$

So, the simplified expression is:

$f(x) \times h(x) = 24x^3 + 184x^2 + 112x$

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4. Find and simplify $\frac{h(x)}{g(x)}$

We are given:

• $g(x) = x + 7$
• $h(x) = 3x^2 + 23x + 14$

To find $\frac{h(x)}{g(x)}$, we need to factor $h(x)$ and check if any common factors can be canceled.

First, factor $h(x) = 3x^2 + 23x + 14$:

Find two numbers that multiply to $3 \times 14 = 42$ and add to 23. These numbers are 21 and 2.

So, we can factor $h(x)$ as follows:

$h(x) = 3x^2 + 21x + 2x + 14$ $h(x) = 3x(x + 7) + 2(x + 7)$ $h(x) = (3x + 2)(x + 7)$

Now, simplify $\frac{h(x)}{g(x)}$:

$\frac{h(x)}{g(x)} = \frac{(3x + 2)(x + 7)}{x + 7}$

Cancel the common factor $(x + 7)$:

$\frac{h(x)}{g(x)} = 3x + 2$

So, the simplified expression is:

$\frac{h(x)}{g(x)} = 3x + 2$

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5. Domain restriction for $\frac{g(x)}{f(x)}$

We are given:

• $g(x) = x + 7$
• $f(x) = 8x$

To find the domain restriction for $\frac{g(x)}{f(x)}$, remember that division by zero is undefined. Set $f(x) = 0$ and solve for $x$:

$8x = 0$ $x = 0$

So, the domain restriction is $x \neq 0$.

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Summary:

1. $g(x) + h(x) = 3x^2 + 24x + 21$
2. $h(x) - g(x) = 3x^2 + 22x + 7$
3. $f(x) \times h(x) = 24x^3 + 184x^2 + 112x$
4. $\frac{h(x)}{g(x)} = 3x + 2$
5. Domain restriction for $\frac{g(x)}{f(x)}$: $x \neq 0$

Would you like further explanation on any of the steps?

Related Questions:

1. How would the results change if $f(x) = 8x^2$?
2. What is the derivative of $g(x) + h(x)$?
3. How would you factor $3x^2 + 24x + 21$?
4. What is the range of $\frac{h(x)}{g(x)}$?
5. How does the product of two functions affect their domains?

Tip:

When factoring quadratic expressions, always check if there's a common factor before applying other factoring techniques.