Question

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

Answer 1

To find the product of \((2x^2 + 3x + 2)(5x^2 - 10x + 3)\) using the distributive property, we will apply the distributive property (also known as the FOIL method for binomials) to each term in the first polynomial with each term in the second polynomial.

Let's break it down step by step:

1. **Distribute \(2x^2\)**:
- \(2x^2 \cdot 5x^2 = 10x^4\)
- \(2x^2 \cdot (-10x) = -20x^3\)
- \(2x^2 \cdot 3 = 6x^2\)

2. **Distribute \(3x\)**:
- \(3x \cdot 5x^2 = 15x^3\)
- \(3x \cdot (-10x) = -30x^2\)
- \(3x \cdot 3 = 9x\)

3. **Distribute \(2\)**:
- \(2 \cdot 5x^2 = 10x^2\)
- \(2 \cdot (-10x) = -20x\)
- \(2 \cdot 3 = 6\)

Now, we will combine all these products:

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

Next, we simplify by combining like terms:

1. For \(x^4\) term: \(10x^4\)
2. For \(x^3\) term: \(-20x^3 + 15x^3 = -5x^3\)
3. For \(x^2\) term: \(6x^2 - 30x^2 + 10x^2 = -14x^2\)
4. For \(x\) term: \(9x - 20x = -11x\)
5. Constant term: \(6\)

Combining everything together, we get:

\[
10x^4 - 5x^3 - 14x^2 - 11x + 6
\]

Thus, the product of \((2x^2 + 3x + 2)(5x^2 - 10x + 3)\) is:

\[
\boxed{10x^4 - 5x^3 - 14x^2 - 11x + 6}
\]