To factor the quadratic expression \( 3g^2 + 4g - 7 \), we can use the method of grouping or the quadratic formula if needed.
First, let's look for two numbers that multiply to \( 3 \times -7 = -21 \) and add to \( 4 \). The factors of \(-21\) that add up to \(4\) are \(7\) and \(-3\).
Next, we can rewrite the middle term \(4g\) using these two numbers:
\[ 3g^2 + 7g - 3g - 7 \]
Now, we can group the terms:
\[ (3g^2 + 7g) + (-3g - 7) \]
Factoring out the common factors in each group gives us:
\[ g(3g + 7) - 1(3g + 7) \]
Now we can factor out the common binomial factor \((3g + 7)\):
\[ (3g + 7)(g - 1) \]
Therefore, the completely factored form of \(3g^2 + 4g - 7\) is:
\[ (3g + 7)(g - 1) \]