Let's simplify the given expression step-by-step:
1. Identify the common denominator:
We have the following expression:
[tex]\[
\frac{g}{g^2 + 6g - 7} + \frac{6}{g - 1}
\][/tex]
We need to combine these terms into a single fraction. First, we factorize the denominator [tex]\(g^2 + 6g - 7\)[/tex]:
[tex]\[
g^2 + 6g - 7 = (g - 1)(g + 7)
\][/tex]
So, the common denominator for our fractions is [tex]\((g - 1)(g + 7)\)[/tex].
2. Rewrite each term with the common denominator:
The first term already has the common denominator:
[tex]\[
\frac{g}{(g - 1)(g + 7)}
\][/tex]
For the second term, we multiply the numerator and the denominator by [tex]\((g + 7)\)[/tex] to achieve the common denominator:
[tex]\[
\frac{6}{g - 1} \cdot \frac{(g + 7)}{(g + 7)} = \frac{6(g + 7)}{(g - 1)(g + 7)}
\][/tex]
3. Combine the terms:
Now we add the fractions:
[tex]\[
\frac{g}{(g - 1)(g + 7)} + \frac{6(g + 7)}{(g - 1)(g + 7)} = \frac{g + 6(g + 7)}{(g - 1)(g + 7)}
\][/tex]
4. Simplify the numerator:
Expand and simplify the numerator:
[tex]\[
g + 6(g + 7) = g + 6g + 42 = 7g + 42
\][/tex]
So we have:
[tex]\[
\frac{7g + 42}{(g - 1)(g + 7)}
\][/tex]
5. Compare with given options:
Let's redefine the simplified expression and compare it with each given option:
[tex]\[
\frac{7g + 42}{(g - 1)(g + 7)} \\
\][/tex]
Evaluating the options:
- Option 1: [tex]\(\frac{7g + 42}{g^2 + 6g - 7}\)[/tex]
Since [tex]\((g^2 + 6g - 7) = (g - 1)(g + 7)\)[/tex], this matches our simplified fraction.
[tex]\[
\boxed{\frac{7g + 42}{(g - 1)(g + 7)}} = \frac{7g + 42}{g^2 + 6g - 7}
\][/tex]
- Option 2: [tex]\(\frac{g^2 + 11g + 34}{g^2 + 6g - 7}\)[/tex]
This expression does not match our simplified fraction because the numerator is different.
- Option 3: [tex]\(\frac{g - 10}{g - 1}\)[/tex]
This expression does not match our simplified fraction because both the numerator and denominator are different.
- Option 4: [tex]\(g + 10\)[/tex]
This expression is not a fraction and hence does not match our simplified fraction.
Therefore, the correct option that represents the simplified form of the given expression is:
[tex]\[
\frac{7g + 42}{g^2 + 6g - 7}
\][/tex]
This matches Option 1. Thus, our simplified expression is:
[tex]\[
\boxed{\frac{7g + 42}{g^2 + 6g - 7}}
\][/tex]
