`(3x^(2)+x-11)xx(7x^(3)+12)`

Answer

To solve the multiplication of the polynomials \( (3x^2 + x - 11) \times (7x^3 + 12) \), we will follow the distributive property (also known as the FOIL method for binomials). Here’s a step-by-step solution:

### Step 1: Identify the polynomials
We have two polynomials:
- \( P(x) = 3x^2 + x - 11 \)
- \( Q(x) = 7x^3 + 12 \)

### Step 2: Distribute each term in the first polynomial by each term in the second polynomial
We will multiply each term in \( P(x) \) by each term in \( Q(x) \).

1. Multiply \( 3x^2 \) by \( 7x^3 \):
\[ 3x^2 \cdot 7x^3 = 21x^{5} \]

2. Multiply \( 3x^2 \) by \( 12 \):
\[ 3x^2 \cdot 12 = 36x^{2} \]

3. Multiply \( x \) by \( 7x^3 \):
\[ x \cdot 7x^3 = 7x^{4} \]

4. Multiply \( x \) by \( 12 \):
\[ x \cdot 12 = 12x \]

5. Multiply \( -11 \) by \( 7x^3 \):
\[ -11 \cdot 7x^3 = -77x^{3} \]

6. Multiply \( -11 \) by \( 12 \):
\[ -11 \cdot 12 = -132 \]

### Step 3: Combine all the results
Now, we will combine all the terms we calculated:
\[ 21x^{5} + 7x^{4} - 77x^{3} + 36x^{2} + 12x - 132 \]

### Step 4: Write the final expression in standard form
We will arrange the terms in descending order of their powers:
\[ 21x^{5} + 7x^{4} - 77x^{3} + 36x^{2} + 12x - 132 \]

### Final Answer
The final expression after multiplying the two polynomials is:
\[ 21x^{5} + 7x^{4} - 77x^{3} + 36x^{2} + 12x - 132 \]

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