I see you've uploaded a photo containing some handwritten mathematical problems. I'll first transcribe and interpret them, then provide solutions.

Transcription of the image:

1. Simplify: $\frac{(a - 3b)^2 - (a + 3b)^2}{(a + b)(a - b)}$

2. Solve: $x^2 - 40x + 25 = 0$

3. Simplify: $\frac{5n + m}{4n^2 + m^2} - \frac{m}{4n^2 + m^2}$

4. Simplify: $\frac{b^2 - 3c - 4}{c^2 - 2}$

5. Simplify: $\frac{bc}{c^2 - 2}$

6. Expressions involving square roots: $\sqrt{81} - \sqrt{16} = ?$ $\sqrt{32} + \sqrt{18} = ?$ $\sqrt{12 \cdot 4} = ?$ $\sqrt{84^2 - 40^2} = ?$

Let's work through these problems:

Problem 1: Simplification

We start with: $\frac{(a - 3b)^2 - (a + 3b)^2}{(a + b)(a - b)}$ The numerator is a difference of squares: [ (a - 3b)^2 - (a + 3b)^2 = \left[(a - 3b) - (a + 3b)\right]\left[(a - 3b) + (a + 3b)\right] ] This simplifies to: $\left(a - 3b - a - 3b\right)\left(a - 3b + a + 3b\right) = (-6b)(2a) = -12ab$ So, the expression becomes: $\frac{-12ab}{(a + b)(a - b)}$ This is the final simplified expression.

Problem 2: Solve the quadratic equation

The equation is: $x^2 - 40x + 25 = 0$ We'll use the quadratic formula: $x = \frac{-(-40) \pm \sqrt{(-40)^2 - 4(1)(25)}}{2(1)}$ Calculating: $x = \frac{40 \pm \sqrt{1600 - 100}}{2} = \frac{40 \pm \sqrt{1500}}{2}$ $x = \frac{40 \pm 10\sqrt{15}}{2} = 20 \pm 5\sqrt{15}$ Thus, the solutions are: $x = 20 + 5\sqrt{15}, \quad x = 20 - 5\sqrt{15}$

Problem 3: Simplify the algebraic expression

We are given: $\frac{5n + m}{4n^2 + m^2} - \frac{m}{4n^2 + m^2}$ Since the denominators are the same, we can subtract the numerators: $\frac{(5n + m) - m}{4n^2 + m^2} = \frac{5n}{4n^2 + m^2}$ This is the simplified result.

Problem 4: Simplify the rational expression

We are given: $\frac{b^2 - 3c - 4}{c^2 - 2}$ This expression cannot be simplified further unless specific values for $b$ and $c$ are provided.

Problem 5: Simplify the second rational expression

We are given: $\frac{bc}{c^2 - 2}$ This expression also cannot be simplified further without additional context.

Problem 6: Simplifying square roots

1. $\sqrt{81} - \sqrt{16} = 9 - 4 = 5$

2. $\sqrt{32} + \sqrt{18} = 4\sqrt{2} + 3\sqrt{2} = 7\sqrt{2}$

3. $\sqrt{12 \cdot 4} = \sqrt{48} = 4\sqrt{3}$

4. $\sqrt{84^2 - 40^2} = \sqrt{7056 - 1600} = \sqrt{5456}$ This can be simplified further as: $\sqrt{5456} = 2\sqrt{1364} = 2\sqrt{4 \cdot 341} = 4\sqrt{341}$ So the simplified form is $4\sqrt{341}$.

Would you like more details on any of these steps, or do you have any further questions?

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