https://file.aiquickdraw.com/mathbot/file/1725536116928qfgv0zoz.jpg

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?

### Related Questions:
1. How do you solve more complex quadratic equations?
2. Can you explain how to simplify expressions involving

Simplify various algebraic expressions, solve a quadratic equation, and work through problems involving square roots.

This webpage covers simplifying algebraic expressions, solving quadratic equations, and handling problems with square roots. Learn step-by-step methods using the quadratic formula, difference of squares, and properties of square roots. Suitable for students in Grades 8-10.

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