To solve the equation [tex]\((3x + 7)(x - 7) = 0\)[/tex] by factoring, we rely on the zero-product property. The zero-product property states that if a product of two factors is zero, then at least one of the factors must be zero. Let's apply these steps to find the solution.
1. Set each factor equal to zero:
[tex]\[
(3x + 7) = 0
\][/tex]
and
[tex]\[
(x - 7) = 0.
\][/tex]
2. Solve each resulting equation separately:
a. For the equation [tex]\(3x + 7 = 0\)[/tex]:
[tex]\[
3x + 7 = 0
\][/tex]
Subtract 7 from both sides:
[tex]\[
3x = -7
\][/tex]
Divide both sides by 3:
[tex]\[
x = -\frac{7}{3}
\][/tex]
b. For the equation [tex]\(x - 7 = 0\)[/tex]:
[tex]\[
x - 7 = 0
\][/tex]
Add 7 to both sides:
[tex]\[
x = 7
\][/tex]
3. Write the solution set:
The solutions to the equation [tex]\((3x + 7)(x - 7) = 0\)[/tex] are:
[tex]\[
x = -\frac{7}{3} \quad \text{and} \quad x = 7.
\][/tex]
Therefore, the solutions are [tex]\(\boxed{-\frac{7}{3}, 7}\)[/tex].
