Question

Expand the expression to a polynomial in standard form:
left parenthesis, 2, x, minus, 3, right parenthesis, left parenthesis, 2, x, plus, 1, right parenthesis, left parenthesis, x, plus, 2, right parenthesis, left parenthesis, x, minus, 2, right parenthesis
(2x−3)(2x+1)(x+2)(x−2)





Solve the problem just like the example below
⬇️
Expand the expression to a polynomial in standard form:
left parenthesis, 2, x, plus, 1, right parenthesis, left parenthesis, 3, x, minus, 5, right parenthesis, left parenthesis, 3, x, plus, 1, right parenthesis
(2x+1)(3x−5)(3x+1)
left parenthesis, 2, x, plus, 1, right parenthesis, left parenthesis, 9, x, squared, minus, 12, x, minus, 5, right parenthesis
(2x+1)(9x
2
−12x−5)

9, x, squared
9x
2
minus, 12, x
−12x
minus, 5
−5
2, x
2x
18, x, cubed
18x
3
minus, 24, x, squared
−24x
2
minus, 10, x
−10x
plus, 1
+1
9, x, squared
9x
2
minus, 12, x
−12x
minus, 5
−5
Combine like terms:
Combine like terms:
18, x, cubed, minus, 15, x, squared, minus, 22, x, minus, 5
18x
3
−15x
2
−22x−5

Answer 1

To expand the expression \((2x-3)(2x+1)(x+2)(x-2)\) to a polynomial in standard form, we can follow these steps:

1. **First, expand \((x+2)(x-2)\)** (using the difference of squares):
\[
(x+2)(x-2) = x^2 - 4
\]

2. **Next, expand \((2x-3)(2x+1)\)**:
\[
(2x-3)(2x+1) = 2x(2x) + 2x(1) - 3(2x) - 3(1) = 4x^2 + 2x - 6x - 3 = 4x^2 - 4x - 3
\]

3. **Now, we can expand \((4x^2 - 4x - 3)(x^2 - 4)\)**:
\[
(4x^2 - 4x - 3)(x^2 - 4)
\]
Using the distributive property (FOIL), we multiply each term in the first polynomial by each term in the second:
\[
= 4x^2 \cdot x^2 + 4x^2 \cdot (-4) - 4x \cdot x^2 - 4x \cdot (-4) - 3 \cdot x^2 - 3 \cdot (-4)
\]
Calculating each term:
\[
= 4x^4 - 16x^2 - 4x^3 + 16x - 3x^2 + 12
\]

4. **Combine like terms**:
Now, we can combine the \(x^2\) terms:
\[
= 4x^4 - 4x^3 + (-16x^2 - 3x^2) + 16x + 12 = 4x^4 - 4x^3 - 19x^2 + 16x + 12
\]

Thus, the polynomial in standard form is:
\[
\boxed{4x^4 - 4x^3 - 19x^2 + 16x + 12}
\]