12) Simply the rational expression $$ \frac{v^{2}+10 v+16}{4 v^{2}+12 v-160} $$

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We start with the expression
$$
\frac{v^{2}+10v+16}{4v^{2}+12v-160}.
$$

**Step 1. Factor the numerator.**

The numerator is
$$
v^{2}+10v+16.
$$
We look for two numbers that multiply to $$16$$ and add to $$10$$. These numbers are $$2$$ and $$8$$. Hence,
$$
v^{2}+10v+16 = (v+2)(v+8).
$$

**Step 2. Factor the denominator.**

The denominator is
$$
4v^{2}+12v-160.
$$
First, factor out the common factor $$4$$:
$$
4v^{2}+12v-160 = 4(v^{2}+3v-40).
$$
Now, factor the quadratic $$v^{2}+3v-40$$. We need two numbers that multiply to $$-40$$ and add to $$3$$. These numbers are $$8$$ and $$-5$$ because:
$$
8 \cdot (-5) = -40 \quad \text{and} \quad 8 + (-5) = 3.
$$
Thus,
$$
v^{2}+3v-40 = (v+8)(v-5).
$$
So the factored denominator is
$$
4(v+8)(v-5).
$$

**Step 3. Simplify the expression.**

Now we have:
$$
\frac{(v+2)(v+8)}{4(v+8)(v-5)}.
$$
Cancel the common factor $$v+8$$ (provided $$v \neq -8$$):
$$
\frac{v+2}{4(v-5)}.
$$

The simplified rational expression is:
$$
\frac{v+2}{4(v-5)}.
$$
The simplified rational expression is $$ \frac{v+2}{4(v-5)} $$.

Simply the rational expression

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