### Instruction: Factor the following expression: $145b^2 +29b$.### Response: I notice that both terms have a common factor of $b$, so I can factor that out: $145b^2 + 29b = b(145b + 29)$.
Now I want to see if I can factor the expression inside the parentheses, $145b + 29$.
I look for the greatest common factor of $145$ and $29$.
I can use the Euclidean algorithm to find it:
Divide $145$ by $29$ and get a quotient of $5$ and a remainder of $0$: $145 = 29 \times 5 + 0$.
Since the remainder is zero, the divisor $29$ is the greatest common factor of $145$ and $29$.
So I can factor out $29$ from $145b + 29$: $145b + 29 = 29(5b + 1)$.
Now I have factored the original expression as: $145b^2 + 29b = b(29)(5b + 1)$.
I check if I can factor any of the factors further, but I see that they are all prime, so I cannot.