Multiply the fothowing and state the restrictions. $$ \frac{t^{2}+19 t+84}{4 t-4} \cdot \frac{2 t-2}{t^{2}+9 t+14} $$

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UpStudy AI · Accepted Answer

To multiply the given fractions and state the restrictions, we will follow these steps:

1. **Factor the expressions where possible.**
2. **Multiply the fractions.**
3. **State the restrictions based on the denominators.**

### Step 1: Factor the expressions

Let's factor each part of the fractions.

1. **Numerator of the first fraction:** $$ t^2 + 19t + 84 $$
   - We need to find two numbers that multiply to $$ 84 $$ and add to $$ 19 $$. These numbers are $$ 12 $$ and $$ 7 $$.
   - Thus, $$ t^2 + 19t + 84 = (t + 12)(t + 7) $$.

2. **Denominator of the first fraction:** $$ 4t - 4 $$
   - This can be factored as $$ 4(t - 1) $$.

3. **Numerator of the second fraction:** $$ 2t - 2 $$
   - This can be factored as $$ 2(t - 1) $$.

4. **Denominator of the second fraction:** $$ t^2 + 9t + 14 $$
   - We need to find two numbers that multiply to $$ 14 $$ and add to $$ 9 $$. These numbers are $$ 7 $$ and $$ 2 $$.
   - Thus, $$ t^2 + 9t + 14 = (t + 7)(t + 2) $$.

Now we can rewrite the expression with the factored forms:

$$
\frac{(t + 12)(t + 7)}{4(t - 1)} \cdot \frac{2(t - 1)}{(t + 7)(t + 2)}
$$

### Step 2: Multiply the fractions

Now we can multiply the fractions:

$$
\frac{(t + 12)(t + 7) \cdot 2(t - 1)}{4(t - 1) \cdot (t + 7)(t + 2)}
$$

Next, we can cancel out the common factors:

- The $$ (t + 7) $$ in the numerator and denominator cancels out.
- The $$ (t - 1) $$ in the numerator and denominator cancels out.

This simplifies to:

$$
\frac{2(t + 12)}{4(t + 2)}
$$

We can further simplify this:

$$
\frac{2(t + 12)}{4(t + 2)} = \frac{1(t + 12)}{2(t + 2)} = \frac{t + 12}{2(t + 2)}
$$

### Step 3: State the restrictions

The restrictions come from the original denominators. We need to ensure that the denominators do not equal zero:

1. From $$ 4(t - 1) $$: $$ t - 1 \neq 0 $$ → $$ t \neq 1 $$
2. From $$ t + 7 $$: $$ t + 7 \neq 0 $$ → $$ t \neq -7 $$
3. From $$ t + 2 $$: $$ t + 2 \neq 0 $$ → $$ t \neq -2 $$

### Final Result

The final simplified expression is:

$$
\frac{t + 12}{2(t + 2)}
$$

And the restrictions are:

- $$ t \neq 1 $$
- $$ t \neq -7 $$
- $$ t \neq -2 $$
The simplified expression is $$ \frac{t + 12}{2(t + 2)} $$ with restrictions $$ t \neq 1 $$, $$ t \neq -7 $$, and $$ t \neq -2 $$.

Multiply the fothowing and state the restrictions. \( \frac{t^{2}+19 t+84}{4 t-4} \cdot \frac{2 t-2}{t^{2}+9 t+14} \)

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