Question

Use distributive property to find the product of
(2𝑥
2
+ 3𝑥 + 2)(5𝑥
2 − 10𝑥 + 3)

Answer 1

To find the product of the polynomials \((2x^2 + 3x + 2)(5x^2 - 10x + 3)\) using the distributive property, we will multiply each term in the first polynomial by each term in the second polynomial.

Let's denote the first polynomial as \(A = 2x^2 + 3x + 2\) and the second polynomial as \(B = 5x^2 - 10x + 3\).

### Step 1: Multiply each term in \(A\) by each term in \(B\)

1. **Multiplying \(2x^2\) by each term in \(B\)**:
- \(2x^2 \cdot 5x^2 = 10x^4\)
- \(2x^2 \cdot (-10x) = -20x^3\)
- \(2x^2 \cdot 3 = 6x^2\)

So, the result of multiplying \(2x^2\) by \(B\) is:
\[
10x^4 - 20x^3 + 6x^2
\]

2. **Multiplying \(3x\) by each term in \(B\)**:
- \(3x \cdot 5x^2 = 15x^3\)
- \(3x \cdot (-10x) = -30x^2\)
- \(3x \cdot 3 = 9x\)

So, the result of multiplying \(3x\) by \(B\) is:
\[
15x^3 - 30x^2 + 9x
\]

3. **Multiplying \(2\) by each term in \(B\)**:
- \(2 \cdot 5x^2 = 10x^2\)
- \(2 \cdot (-10x) = -20x\)
- \(2 \cdot 3 = 6\)

So, the result of multiplying \(2\) by \(B\) is:
\[
10x^2 - 20x + 6
\]

### Step 2: Combine all results
Now add all the results together:

\[
(10x^4 - 20x^3 + 6x^2) + (15x^3 - 30x^2 + 9x) + (10x^2 - 20x + 6)
\]

### Step 3: Group like terms
Now, let's group the like terms:
- For \(x^4\): \(10x^4\)
- For \(x^3\): \(-20x^3 + 15x^3 = -5x^3\)
- For \(x^2\): \(6x^2 - 30x^2 + 10x^2 = -14x^2\)
- For \(x\): \(9x - 20x = -11x\)
- Constant: \(6\)

### Step 4: Write the final result
Putting it all together, we have:
\[
10x^4 - 5x^3 - 14x^2 - 11x + 6
\]

So, the product of \((2x^2 + 3x + 2)(5x^2 - 10x + 3)\) is:
\[
\boxed{10x^4 - 5x^3 - 14x^2 - 11x + 6}
\]