Find each of the following products :
`(9x^(2) - x + 15) xx (x^(2) - x - 1)`

Answer

To find the product of the algebraic expressions \( (9x^{2} - x + 15) \) and \( (x^{2} - x - 1) \), we will use the distributive property (also known as the FOIL method for binomials). Here’s the step-by-step solution:

### Step 1: Distribute \( 9x^{2} \)
We will multiply \( 9x^{2} \) with each term in the second expression \( (x^{2} - x - 1) \).

\[ 9x^{2} \cdot x^{2} = 9x^{4} \]
\[ 9x^{2} \cdot (-x) = -9x^{3} \]
\[ 9x^{2} \cdot (-1) = -9x^{2} \]

### Step 2: Distribute \( -x \)
Next, we will multiply \( -x \) with each term in the second expression.

\[ -x \cdot x^{2} = -x^{3} \]
\[ -x \cdot (-x) = x^{2} \]
\[ -x \cdot (-1) = x \]

### Step 3: Distribute \( 15 \)
Now, we will multiply \( 15 \) with each term in the second expression.

\[ 15 \cdot x^{2} = 15x^{2} \]
\[ 15 \cdot (-x) = -15x \]
\[ 15 \cdot (-1) = -15 \]

### Step 4: Combine all the results
Now, we will combine all the results from the distributions:

\[ 9x^{4} + (-9x^{3}) + (-x^{3}) + (-9x^{2}) + x^{2} + 15x^{2} + x + (-15) \]

### Step 5: Combine like terms
Now, we will combine the like terms:

- For \( x^{4} \): \( 9x^{4} \)
- For \( x^{3} \): \( -9x^{3} - x^{3} = -10x^{3} \)
- For \( x^{2} \): \( -9x^{2} + x^{2} + 15x^{2} = 7x^{2} \)
- For \( x \): \( x - 15x = -14x \)
- Constant term: \( -15 \)

### Final Result
Putting it all together, we get:

\[ 9x^{4} - 10x^{3} + 7x^{2} - 14x - 15 \]