7. $$ x^{4}-x^{2} $$ $$ 2 \cdot-a^{4}+16 $$ $$ 3 \cdot \frac{1}{4 b^{2}-\frac{4}{16}} $$ $$ 4 \cdot(a+x)^{2}-y^{2} $$ $$ 5 \cdot 81 x^{2}-36 $$ $$ 6 \cdot 10 b(x-y)+5 $$ 7. $$ t^{2}(4-t)+4(4-t) $$ 8. $$ 12 x y^{2}+4 x y-3 x^{3} y^{3} $$

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Sure! Let's go through each of the given expressions and factor or simplify them step by step.

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### 1. $$ x^{4} - x^{2} $$

**Factorization:**
$$
x^{4} - x^{2} = x^{2}(x^{2} - 1) = x^{2}(x - 1)(x + 1)
$$

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### 2. $$ -a^{4} + 16 $$

**Simplification and Factorization:**
$$
- a^{4} + 16 = 16 - a^{4}
$$
This is a difference of squares:
$$
16 - a^{4} = (4)^2 - (a^{2})^2 = (4 - a^{2})(4 + a^{2})
$$
Further factoring $$4 - a^{2}$$:
$$
4 - a^{2} = (2 - a)(2 + a)
$$
**Final Factored Form:**
$$
(2 - a)(2 + a)(4 + a^{2})
$$

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### 3. $$ 3 \cdot \frac{1}{4 b^{2} - \frac{4}{16}} $$

**Simplification:**
$$
3 \cdot \frac{1}{4b^{2} - \frac{1}{4}} = \frac{3}{4b^{2} - \frac{1}{4}} = \frac{3}{\frac{16b^{2} - 1}{4}} = \frac{12}{16b^{2} - 1}
$$
Factor the denominator as a difference of squares:
$$
16b^{2} - 1 = (4b)^{2} - 1^{2} = (4b - 1)(4b + 1)
$$
**Final Simplified Form:**
$$
\frac{12}{(4b - 1)(4b + 1)}
$$

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### 4. $$ (a + x)^{2} - y^{2} $$

**Factorization:**
This is a difference of squares:
$$
(a + x)^{2} - y^{2} = (a + x - y)(a + x + y)
$$

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### 5. $$ 81x^{2} - 36 $$

**Factorization:**
$$
81x^{2} - 36 = 9(9x^{2} - 4) = 9(3x)^{2} - 2^{2} = 9(3x - 2)(3x + 2)
$$
**Final Factored Form:**
$$
9(3x - 2)(3x + 2)
$$

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### 6. $$ 10b(x - y) + 5 $$

**Factorization:**
Factor out the greatest common factor, which is 5:
$$
10b(x - y) + 5 = 5[2b(x - y) + 1]
$$
**Final Factored Form:**
$$
5[2b(x - y) + 1]
$$

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### 7. $$ t^{2}(4 - t) + 4(4 - t) $$

**Factorization:**
Factor out the common term $$(4 - t)$$:
$$
t^{2}(4 - t) + 4(4 - t) = (4 - t)(t^{2} + 4)
$$
**Final Factored Form:**
$$
(4 - t)(t^{2} + 4)
$$

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### 8. $$ 12xy^{2} + 4xy - 3x^{3}y^{3} $$

**Factorization:**
1. Factor out the greatest common factor, which is $$xy$$:
$$
12xy^{2} + 4xy - 3x^{3}y^{3} = xy(12y + 4 - 3x^{2}y^{2})
$$
2. Factor out a 4 from the first two terms inside the parentheses:
$$
xy(4(3y + 1) - 3x^{2}y^{2})
$$
**Final Factored Form:**
$$
xy[4(3y + 1) - 3x^{2}y^{2}]
$$
*Note:* Further factoring isn't straightforward unless additional relationships between variables are provided.

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Feel free to ask if you need further clarification on any of these steps!
Here are the simplified forms of the given expressions: 1. $$ x^{4} - x^{2} = x^{2}(x - 1)(x + 1) $$ 2. $$ -a^{4} + 16 = (2 - a)(2 + a)(4 + a^{2}) $$ 3. $$ 3 \cdot \frac{1}{4b^{2} - \frac{1}{4}} = \frac{12}{(4b - 1)(4b + 1)} $$ 4. $$ (a + x)^{2} - y^{2} = (a + x - y)(a + x + y) $$ 5. $$ 81x^{2} - 36 = 9(3x - 2)(3x + 2) $$ 6. $$ 10b(x - y) + 5 = 5[2b(x - y) + 1] $$ 7. $$ t^{2}(4 - t) + 4(4 - t) = (4 - t)(t^{2} + 4) $$ 8. $$ 12xy^{2} + 4xy - 3x^{3}y^{3} = xy[4(3y + 1) - 3x^{2}y^{2}] $$

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