Multiply `(5x^(2) - 6x + 9)` by (2x - 3)

Answer

To solve the problem of multiplying the algebraic expression \( (5x^2 - 6x + 9) \) by \( (2x - 3) \), we will follow these steps:

### Step-by-Step Solution:

1. **Distribute \( (2x - 3) \) to each term in \( (5x^2 - 6x + 9) \)**:
\[ (2x - 3)(5x^2 - 6x + 9) = 2x(5x^2) + 2x(-6x) + 2x(9) - 3(5x^2) - 3(-6x) - 3(9) \]

2. **Multiply each term**:
- For \( 2x \cdot 5x^2 \):
\[ 2 \cdot 5 = 10 \quad \text{and} \quad x \cdot x^2 = x^{1+2} = x^3 \quad \Rightarrow \quad 10x^3 \]
- For \( 2x \cdot -6x \):
\[ 2 \cdot -6 = -12 \quad \text{and} \quad x \cdot x = x^{1+1} = x^2 \quad \Rightarrow \quad -12x^2 \]
- For \( 2x \cdot 9 \):
\[ 2 \cdot 9 = 18 \quad \Rightarrow \quad 18x \]
- For \( -3 \cdot 5x^2 \):
\[ -3 \cdot 5 = -15 \quad \Rightarrow \quad -15x^2 \]
- For \( -3 \cdot -6x \):
\[ -3 \cdot -6 = 18 \quad \Rightarrow \quad 18x \]
- For \( -3 \cdot 9 \):
\[ -3 \cdot 9 = -27 \]

3. **Combine all the terms**:
\[ 10x^3 + (-12x^2 - 15x^2) + (18x + 18x) - 27 \]

4. **Combine like terms**:
- Combine \( -12x^2 \) and \( -15x^2 \):
\[ -12x^2 - 15x^2 = -27x^2 \]
- Combine \( 18x \) and \( 18x \):
\[ 18x + 18x = 36x \]

5. **Final expression**:
\[ 10x^3 - 27x^2 + 36x - 27 \]

### Final Answer:
\[ 10x^3 - 27x^2 + 36x - 27 \]