To factor the expression \( v^2 + 8vg + 7g^2 \), we can look for two numbers that multiply to \( 7g^2 \) (the constant term) and add up to \( 8g \) (the coefficient of the middle term).
The factors of \( 7g^2 \) that add up to \( 8g \) are \( 7g \) and \( 1g \):
- \( 7g \times 1g = 7g^2 \)
- \( 7g + 1g = 8g \)
Now we can rewrite the expression in factored form:
\[ v^2 + 8vg + 7g^2 = (v + 7g)(v + 1g) \]
Thus, the completely factored form of the expression \( v^2 + 8vg + 7g^2 \) is:
\[ (v + 7g)(v + g) \]