Question

Expand the expression to a polynomial in standard form:
left parenthesis, 4, x, squared, plus, 3, x, minus, 3, right parenthesis, left parenthesis, 2, x, squared, plus, 3, x, plus, 3, right parenthesis
(4x
2
+3x−3)(2x
2
+3x+3)

Answer 1

To expand the expression \((4x^2 + 3x - 3)(2x^2 + 3x + 3)\), we can use the distributive property (also known as the FOIL method for binomials, but here it’s applied to polynomials).

We will multiply each term in the first polynomial by each term in the second polynomial:

1. **Multiply \(4x^2\) by each term in the second polynomial:**
- \(4x^2 \cdot 2x^2 = 8x^4\)
- \(4x^2 \cdot 3x = 12x^3\)
- \(4x^2 \cdot 3 = 12x^2\)

2. **Multiply \(3x\) by each term in the second polynomial:**
- \(3x \cdot 2x^2 = 6x^3\)
- \(3x \cdot 3x = 9x^2\)
- \(3x \cdot 3 = 9x\)

3. **Multiply \(-3\) by each term in the second polynomial:**
- \(-3 \cdot 2x^2 = -6x^2\)
- \(-3 \cdot 3x = -9x\)
- \(-3 \cdot 3 = -9\)

Now we can combine all of these results:

\[
8x^4 + 12x^3 + 12x^2 + 6x^3 + 9x^2 + 9x - 6x^2 - 9x - 9
\]

Next, we combine like terms:

- The \(x^4\) term: \(8x^4\)
- The \(x^3\) terms: \(12x^3 + 6x^3 = 18x^3\)
- The \(x^2\) terms: \(12x^2 + 9x^2 - 6x^2 = 15x^2\)
- The \(x\) terms: \(9x - 9x = 0\)
- The constant term: \(-9\)

Putting it all together, we have:

\[
8x^4 + 18x^3 + 15x^2 - 9
\]

Thus, the expanded polynomial in standard form is:

\[
\boxed{8x^4 + 18x^3 + 15x^2 - 9}
\]