To factor the given perfect square trinomial $25y^2 + 70yz + 49z^2$, we can use the fact that a perfect square trinomial of the form $a^2 + 2ab + b^2$ can be factored as $(a + b)^2$.

Let's identify the components $a$ and $b$ in the given trinomial $25y^2 + 70yz + 49z^2$:

1. $25y^2$ is a perfect square, and $\sqrt{25y^2} = 5y$.
2. $49z^2$ is a perfect square, and $\sqrt{49z^2} = 7z$.
3. The middle term $70yz$ can be written as $2 \times 5y \times 7z$.

Since $25y^2 + 70yz + 49z^2$ fits the pattern $a^2 + 2ab + b^2$ with $a = 5y$ and $b = 7z$, we can factor it as:

$(5y + 7z)^2$

So, the factor of the given perfect square trinomial is:

$(5y + 7z)^2$

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. How do you recognize a perfect square trinomial?
2. Can you factor $16x^2 + 24xy + 9y^2$ in a similar way?
3. What happens if the middle term is negative, for example, $25y^2 - 70yz + 49z^2$?
4. How would you factor a trinomial that is not a perfect square?
5. Can you verify the factorization of $25y^2 + 70yz + 49z^2$ by expanding $(5y + 7z)^2$?
6. What are some real-world applications of factoring trinomials?
7. How do you solve equations involving perfect square trinomials?
8. How would you factor a perfect square trinomial with fractional coefficients?

Tip: When factoring trinomials, always check if it can be expressed as a perfect square trinomial first. This simplifies the process significantly.