https://file.aiquickdraw.com/mathbot/file/17307623935560w95xy66.jpg

Let's go through each expression in your image one by one.

### 1. \((x^4 + 3x^2 - x - 5) + (x^3 - 7x^2 + 9x - 12)\)

To simplify this, we combine like terms:

\[
x^4 + 3x^2 - x - 5 + x^3 - 7x^2 + 9x - 12
\]

Grouping terms by power of \( x \):

\[
= x^4 + x^3 + (3x^2 - 7x^2) + (-x + 9x) + (-5 - 12)
\]

\[
= x^4 + x^3 - 4x^2 + 8x - 17
\]

So, the result is:

\[
x^4 + x^3 - 4x^2 + 8x - 17
\]

---

### 2. \((2x - 3)(3x + 5)\)

To expand this, we apply the distributive property (FOIL method):

\[
(2x)(3x) + (2x)(5) - (3)(3x) - (3)(5)
\]

\[
= 6x^2 + 10x - 9x - 15
\]

\[
= 6x^2 + x - 15
\]

So, the result is:

\[
6x^2 + x - 15
\]

---

### 3. \(3(2x^3 - 3x^2 + 2x - 7)\)

Distribute the \( 3 \) to each term inside the parentheses:

\[
= 3 \cdot 2x^3 - 3 \cdot 3x^2 + 3 \cdot 2x - 3 \cdot 7
\]

\[
= 6x^3 - 9x^2 + 6x - 21
\]

So, the result is:

\[
6x^3 - 9x^2 + 6x - 21
\]

---

### 4. \((2x^3 + 3x^2 - 7x + 5) - (4x^3 - 5x^2 + 3x - 3)\)

Here, we subtract the polynomials term by term:

\[
= (2x^3 - 4x^3) + (3x^2 + 5x^2) + (-7x - 3x) + (5 + 3)
\]

\[
= -2x^3 + 8x^2 - 10x + 8
\]

So, the result is:

\[
-2x^3 + 8x^2 - 10x + 8
\]

---

### 5. \((3x^2 + 2y^3)(3x^2 - 2y^3)\)

This is a difference of squares, so we apply the formula \((a + b)(a - b) = a^2 - b^2\):

\[
= (3x^2)^2 - (2y^3)^2
\]

\[
= 9x^4 - 4y^6
\]

So, the result is:

\[
9x^4 - 4y^6
\]

---

### 6. \(\left(\frac{1}{2} + \frac{5}{3x} + \frac{1}{5x^2}\right) + \left(\frac{7}{2} - \frac{2}{3x} + \frac{4}{5x^2}\right)\)

To simplify, we add each term by finding a common denominator where needed:

\[
= \left(\frac{1}{2} + \frac{7}{2}\right) + \left(\frac{5}{3x} - \frac{2}{3x}\right) + \left(\frac{1}{5x^2} + \frac{4}{5x^2}\right)
\]

\[
= \frac{8}{2} + \frac{3}{3x} + \frac{5}{5x^2}
\]

Simplifying each fraction:

\[
= 4 + \frac{1}{x} + \frac{1}{x^2}
\]

So, the result is:

\[
4 + \frac{1}{x} + \frac{1}{x^2}
\]

---

### Summary of Results

1. \( x^4 + x^3 - 4x^2 + 8x - 17 \)
2. \( 6x^2 + x - 15 \)
3. \( 6x^3 - 9x^2 + 6x - 21 \)
4. \( -2x^3 + 8x^2 - 10x + 8 \)
5. \( 9x^4 - 4y^6 \)
6. \( 4 + \frac{1}{x} + \frac{1}{x^2} \)

Would you like any further details on these solutions?

---

### Related Questions

1. How do you perform polynomial addition and subtraction with multiple terms?
2. What is the difference of squares formula, and when is it used?
3. How do you handle polynomial multiplication using the distributive property?
4. What are some techniques for simplifying complex fractions?
5. How do you find a common denominator when adding fractions with variables?

### Tip
When adding or subtracting polynomials, it’s helpful to align terms by power to make it easier to combine like terms.

Solve the following algebraic expressions: (x^4 + 3x^2 - x - 5) + (x^3 - 7x^2 + 9x - 12), (2x - 3)(3x + 5), 3(2x^3 - 3x^2 + 2x - 7), (2x^3 + 3x^2 - 7x + 5) - (4x^3 - 5x^2 + 3x - 3), (3x^2 + 2y^3)(3x^2 - 2y^3), (1/2 + 5/3x + 1/5x^2) + (7/2 - 2/3x + 4/5x^2)

Detailed solutions for algebraic expressions involving polynomial addition, subtraction, multiplication, and fraction simplification. Learn how to combine like terms, apply the difference of squares, and simplify complex fractions. Suitable for high school students (Grades 10-12).

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