To factor the expression \( 3z(2z+9) + 6(2z+9)^2 \), we first identify the greatest common factor (GCF) of the two terms.
- The first term is \( 3z(2z + 9) \).
- The second term is \( 6(2z + 9)^2 \).
The common factor in both terms is \( (2z + 9) \).
Now, let's express each term in terms of this common factor:
- The first term, \( 3z(2z + 9) \), can be rewritten as \( (2z + 9)(3z) \).
- The second term, \( 6(2z + 9)^2 \), can be rewritten as \( (2z + 9)(6(2z + 9)) \).
Now we can factor out \( (2z + 9) \): \[ 3z(2z + 9) + 6(2z + 9)^2 = (2z + 9)(3z + 6(2z + 9)) \]
Now, simplify the expression inside the parentheses: \[ 3z + 6(2z + 9) = 3z + 12z + 54 = 15z + 54 \]
Thus, we can rewrite the factored expression as: \[ (2z + 9)(15z + 54) \]
Finally, we can factor out a common factor from \( 15z + 54 \). The GCF of \( 15 \) and \( 54 \) is \( 3 \): \[ 15z + 54 = 3(5z + 18) \]
Putting it all together, we have the final completely factored form: \[ 3(2z + 9)(5z + 18) \]
So, the fully factored expression is \( \boxed{3(2z + 9)(5z + 18)} \).