Find each of the following products :
`(x^(3) - 2x^(2) + 5) xx (4x - 1)`

Answer

To find the product of the algebraic expressions \( (x^{3} - 2x^{2} + 5) \) and \( (4x - 1) \), we will use the distributive property (also known as the FOIL method for binomials). Here are the steps:

### Step 1: Distribute \( 4x \) to each term in the trinomial
We start by multiplying \( 4x \) with each term in \( (x^{3} - 2x^{2} + 5) \).

\[ 4x \cdot x^{3} = 4x^{4} \]
\[ 4x \cdot (-2x^{2}) = -8x^{3} \]
\[ 4x \cdot 5 = 20x \]

### Step 2: Distribute \( -1 \) to each term in the trinomial
Next, we multiply \( -1 \) with each term in \( (x^{3} - 2x^{2} + 5) \).

\[ -1 \cdot x^{3} = -x^{3} \]
\[ -1 \cdot (-2x^{2}) = 2x^{2} \]
\[ -1 \cdot 5 = -5 \]

### Step 3: Combine all the results
Now, we combine all the results from Steps 1 and 2.

\[ 4x^{4} - 8x^{3} + 20x - x^{3} + 2x^{2} - 5 \]

### Step 4: Combine like terms
Now, we will combine the like terms:

- The \( x^{3} \) terms: \( -8x^{3} - x^{3} = -9x^{3} \)
- The \( x^{2} \) term: \( 2x^{2} \)
- The \( x \) term: \( 20x \)
- The constant term: \( -5 \)

So, we have:

\[ 4x^{4} - 9x^{3} + 2x^{2} + 20x - 5 \]

### Final Answer
Thus, the product of \( (x^{3} - 2x^{2} + 5) \) and \( (4x - 1) \) is:

\[ 4x^{4} - 9x^{3} + 2x^{2} + 20x - 5 \]

---