Find each of the following products :
`(9x^(2) - x + 15) xx (x^(2) - 3)`

Answer

To find the product of the algebraic expressions \( (9x^2 - x + 15) \) and \( (x^2 - 3) \), we will use the distributive property (also known as the FOIL method for binomials). Here’s the step-by-step solution:

### Step 1: Distribute the first term of the first expression
Multiply \( 9x^2 \) by each term in the second expression:
\[ 9x^2 \cdot (x^2 - 3) = 9x^2 \cdot x^2 - 9x^2 \cdot 3 = 9x^4 - 27x^2 \]

### Step 2: Distribute the second term of the first expression
Multiply \( -x \) by each term in the second expression:
\[ -x \cdot (x^2 - 3) = -x \cdot x^2 + x \cdot 3 = -x^3 + 3x \]

### Step 3: Distribute the third term of the first expression
Multiply \( 15 \) by each term in the second expression:
\[ 15 \cdot (x^2 - 3) = 15 \cdot x^2 - 15 \cdot 3 = 15x^2 - 45 \]

### Step 4: Combine all the results
Now, we will combine all the results from the previous steps:
\[ 9x^4 - 27x^2 - x^3 + 3x + 15x^2 - 45 \]

### Step 5: Combine like terms
Now, we will combine the like terms:
- The \( x^4 \) term: \( 9x^4 \)
- The \( x^3 \) term: \( -x^3 \)
- The \( x^2 \) terms: \( -27x^2 + 15x^2 = -12x^2 \)
- The \( x \) term: \( 3x \)
- The constant term: \( -45 \)

Putting it all together, we have:
\[ 9x^4 - x^3 - 12x^2 + 3x - 45 \]

### Final Answer:
The product of the expressions \( (9x^2 - x + 15) \) and \( (x^2 - 3) \) is:
\[ \boxed{9x^4 - x^3 - 12x^2 + 3x - 45} \]