The provided expression is:

$(7z^2)^2 - (6z^2 + 11^2)^2$

Let's simplify step-by-step.

Step 1: Expand the squares

1. First term: $(7z^2)^2 = 49z^4$

2. Second term: $(6z^2 + 11^2)^2 = (6z^2 + 121)^2$ Expanding this using the binomial theorem: $(6z^2 + 121)^2 = (6z^2)^2 + 2(6z^2)(121) + 121^2$ $= 36z^4 + 1452z^2 + 14641$

Step 2: Subtract the two expressions

$49z^4 - (36z^4 + 1452z^2 + 14641)$

Distribute the negative sign: $49z^4 - 36z^4 - 1452z^2 - 14641$

Combine like terms: $(49z^4 - 36z^4) - 1452z^2 - 14641$ $13z^4 - 1452z^2 - 14641$

Final Answer:

$13z^4 - 1452z^2 - 14641$

Would you like a detailed explanation of any step?

Here are 5 related questions for practice:

1. Can you simplify $(5z)^2 - (3z + 4)^2$?
2. How do you factorize $13z^4 - 1452z^2 - 14641$?
3. What is the general formula for expanding $(a + b)^2$?
4. How do you find the roots of a quartic polynomial like $13z^4 - 1452z^2 - 14641 = 0$?
5. How do terms of different degrees affect the graph of a polynomial?

Tip: Always check for common factors before attempting complex operations like expansions or factorization!