Find each of the following products :
`(3x^(2) + 5x - 9) xx (3x - 5)`

Answer

To find the product of the expressions \( (3x^2 + 5x - 9) \) and \( (3x - 5) \), we will use the distributive property (also known as the FOIL method for binomials). Here’s a step-by-step solution:

### Step 1: Write the expressions
We start with the expressions:
\[ (3x^2 + 5x - 9) \times (3x - 5) \]

### Step 2: Distribute each term in the first expression to each term in the second expression
We will multiply each term in the first polynomial by each term in the second polynomial.

1. **Multiply \(3x^2\) by \(3x\)**:
\[ 3x^2 \times 3x = 9x^3 \]

2. **Multiply \(3x^2\) by \(-5\)**:
\[ 3x^2 \times -5 = -15x^2 \]

3. **Multiply \(5x\) by \(3x\)**:
\[ 5x \times 3x = 15x^2 \]

4. **Multiply \(5x\) by \(-5\)**:
\[ 5x \times -5 = -25x \]

5. **Multiply \(-9\) by \(3x\)**:
\[ -9 \times 3x = -27x \]

6. **Multiply \(-9\) by \(-5\)**:
\[ -9 \times -5 = 45 \]

### Step 3: Combine all the results
Now, we combine all the results from the multiplications:
\[ 9x^3 + (-15x^2 + 15x^2) + (-25x - 27x) + 45 \]

### Step 4: Simplify the expression
1. The \(x^2\) terms cancel each other out:
\[ -15x^2 + 15x^2 = 0 \]

2. Combine the \(x\) terms:
\[ -25x - 27x = -52x \]

So, the final expression is:
\[ 9x^3 - 52x + 45 \]

### Final Answer
The product of \( (3x^2 + 5x - 9) \) and \( (3x - 5) \) is:
\[ 9x^3 - 52x + 45 \]

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