Multiply `2x^(3) - 7x + 8` by `3x^(2) + 2x`.

Answer

To multiply the algebraic expressions \(2x^3 - 7x + 8\) and \(3x^2 + 2x\), we will follow these steps:

### Step 1: Distribute \(3x^2\) to each term in the first expression
We will multiply \(3x^2\) by each term in \(2x^3 - 7x + 8\).

\[ 3x^2 \cdot 2x^3 = 6x^5 \]
\[ 3x^2 \cdot (-7x) = -21x^3 \]
\[ 3x^2 \cdot 8 = 24x^2 \]

So, after distributing \(3x^2\), we have:

\[ 6x^5 - 21x^3 + 24x^2 \]

### Step 2: Distribute \(2x\) to each term in the first expression
Next, we will multiply \(2x\) by each term in \(2x^3 - 7x + 8\).

\[ 2x \cdot 2x^3 = 4x^4 \]
\[ 2x \cdot (-7x) = -14x^2 \]
\[ 2x \cdot 8 = 16x \]

So, after distributing \(2x\), we have:

\[ 4x^4 - 14x^2 + 16x \]

### Step 3: Combine all the results
Now we will combine all the terms from both distributions:

\[ 6x^5 + 4x^4 - 21x^3 + 24x^2 - 14x^2 + 16x \]

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

- The \(x^2\) terms: \(24x^2 - 14x^2 = 10x^2\)

So, the final expression will be:

\[ 6x^5 + 4x^4 - 21x^3 + 10x^2 + 16x \]

### Final Answer:
The product of \(2x^3 - 7x + 8\) and \(3x^2 + 2x\) is:

\[ 6x^5 + 4x^4 - 21x^3 + 10x^2 + 16x \]

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