To solve the equation [tex]\((x + 2)(x - 5)(2x + 7) = 0\)[/tex], we need to find the values of [tex]\(x\)[/tex] that make the equation true. This type of equation is a product of factors that equals zero. According to the zero-product property, if a product of several factors is zero, at least one of the factors must be zero. Therefore, we can set each factor equal to zero and solve for [tex]\(x\)[/tex].
Let's set each factor to zero and solve them one at a time:
1. First factor: [tex]\(x + 2 = 0\)[/tex]
[tex]\[
x + 2 = 0
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
x = -2
\][/tex]
2. Second factor: [tex]\(x - 5 = 0\)[/tex]
[tex]\[
x - 5 = 0
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
x = 5
\][/tex]
3. Third factor: [tex]\(2x + 7 = 0\)[/tex]
[tex]\[
2x + 7 = 0
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
2x = -7
\][/tex]
Dividing both sides by 2:
[tex]\[
x = -\frac{7}{2}
\][/tex]
So, the solutions to the equation [tex]\((x + 2)(x - 5)(2x + 7) = 0\)[/tex] are:
[tex]\[
x = -\frac{7}{2}, \, x = -2, \, \text{and} \, x = 5
\][/tex]
Therefore, the values of [tex]\(x\)[/tex] that satisfy the equation are [tex]\(x = -\frac{7}{2}\)[/tex], [tex]\(x = -2\)[/tex], and [tex]\(x = 5\)[/tex].
