Multiply :
`6x^(3) - 5x + 10` by `4 - 3x^(2)`

Answer

To multiply the algebraic expressions \(6x^3 - 5x + 10\) and \(4 - 3x^2\), we will follow these steps:

### Step 1: Write down the expressions
We have two expressions:
1. \(6x^3 - 5x + 10\)
2. \(4 - 3x^2\)

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

#### Multiplying \(6x^3\):
- \(6x^3 \cdot 4 = 24x^3\)
- \(6x^3 \cdot (-3x^2) = -18x^5\)

#### Multiplying \(-5x\):
- \(-5x \cdot 4 = -20x\)
- \(-5x \cdot (-3x^2) = 15x^3\)

#### Multiplying \(10\):
- \(10 \cdot 4 = 40\)
- \(10 \cdot (-3x^2) = -30x^2\)

### Step 3: Combine all the results
Now we combine all the products we calculated:
- From \(6x^3\): \(24x^3 - 18x^5\)
- From \(-5x\): \(-20x + 15x^3\)
- From \(10\): \(40 - 30x^2\)

Putting it all together:
\[ -18x^5 + (24x^3 + 15x^3) - 30x^2 - 20x + 40 \]

### Step 4: Combine like terms
Now we will combine the like terms:
- The \(x^5\) term: \(-18x^5\)
- The \(x^3\) terms: \(24x^3 + 15x^3 = 39x^3\)
- The \(x^2\) term: \(-30x^2\)
- The \(x\) term: \(-20x\)
- The constant term: \(40\)

So, the final expression is:
\[ -18x^5 + 39x^3 - 30x^2 - 20x + 40 \]

### Final Answer
\[ -18x^5 + 39x^3 - 30x^2 - 20x + 40 \]

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