To find the ratio g(a) / h(a), where g(a) = 3a + 1 and h(a) = 4a + 2, we'll begin by writing down the ratio as a fraction:
\[ \frac{g(a)}{h(a)} = \frac{3a + 1}{4a + 2} \] To simplify this expression, we look for common factors in the numerator and the denominator that we can cancel.
In this case, the denominator has a common factor of 2.
We can factor this out to see if any simplification is possible:
\[ 4a + 2 = 2(2a + 1) \] Now we rewrite the original ratio using this factored form of the denominator:
\[ \frac{g(a)}{h(a)} = \frac{3a + 1}{2(2a + 1)} \] You can see that there is no common factor between the numerator and the factored denominator, which means that this fraction cannot be simplified further.
Thus, the simplified expression for g(a) / h(a) is:
\[ \text{The ratio } \frac{g(a)}{h(a)} = \frac{3a + 1}{2(2a + 1)} \]
This is the simplest form of the ratio g(a) / h(a) as a function of "a".