To solve the given equation [tex]\((2x + 6)(x - 2) = 0\)[/tex], we need to find the values of [tex]\(x\)[/tex] that make the equation true.
1. Understand the equation: The equation is a product of two factors, [tex]\((2x + 6)\)[/tex] and [tex]\((x - 2)\)[/tex], which equals zero. According to the Zero Product Property, if the product of two expressions is zero, then at least one of the expressions must be zero.
2. Set each factor to zero:
- First factor: [tex]\(2x + 6 = 0\)[/tex]
- Second factor: [tex]\(x - 2 = 0\)[/tex]
3. Solve each equation separately:
- For the first factor [tex]\(2x + 6 = 0\)[/tex]:
[tex]\[
2x + 6 = 0
\][/tex]
Subtract 6 from both sides:
[tex]\[
2x = -6
\][/tex]
Divide both sides by 2:
[tex]\[
x = -3
\][/tex]
- For the second factor [tex]\(x - 2 = 0\)[/tex]:
[tex]\[
x - 2 = 0
\][/tex]
Add 2 to both sides:
[tex]\[
x = 2
\][/tex]
4. Present the solutions: The solutions to the equation [tex]\((2x + 6)(x - 2) = 0\)[/tex] are [tex]\(x = -3\)[/tex] and [tex]\(x = 2\)[/tex].
Thus, the correct answer is:
[tex]\[ x = -3 \text{ or } 2 \][/tex]
