To solve the equation [tex]\((p - 7)(p + 3) = -9\)[/tex], follow these steps:
1. Expand the left-hand side:
First, let's expand the product [tex]\((p - 7)(p + 3)\)[/tex].
[tex]\[
(p - 7)(p + 3) = p(p + 3) - 7(p + 3)
\][/tex]
Simplifying this, we get:
[tex]\[
= p^2 + 3p - 7p - 21
\][/tex]
Combining like terms:
[tex]\[
= p^2 - 4p - 21
\][/tex]
2. Set the equation equal to [tex]\(-9\)[/tex]:
Now, we set this expanded equation equal to [tex]\(-9\)[/tex]:
[tex]\[
p^2 - 4p - 21 = -9
\][/tex]
3. Move all terms to one side to set the equation to 0:
Add 9 to both sides to get a standard quadratic equation:
[tex]\[
p^2 - 4p - 21 + 9 = 0
\][/tex]
Simplifying further:
[tex]\[
p^2 - 4p - 12 = 0
\][/tex]
4. Factor the quadratic equation:
Next, we need to factor the quadratic equation [tex]\(p^2 - 4p - 12\)[/tex]. We look for two numbers that multiply to [tex]\(-12\)[/tex] and add to [tex]\(-4\)[/tex]. These numbers are [tex]\(-6\)[/tex] and [tex]\(2\)[/tex]:
[tex]\[
p^2 - 4p - 12 = (p - 6)(p + 2) = 0
\][/tex]
5. Solve for [tex]\(p\)[/tex]:
Set each factor to zero and solve for [tex]\(p\)[/tex]:
[tex]\[
p - 6 = 0 \quad \text{or} \quad p + 2 = 0
\][/tex]
Solving these:
[tex]\[
p = 6 \quad \text{or} \quad p = -2
\][/tex]
Therefore, the solutions to the equation [tex]\((p - 7)(p + 3) = -9\)[/tex] are [tex]\(p = 6\)[/tex] and [tex]\(p = -2\)[/tex].
Thus, the final answer is:
[tex]\[
\boxed{6, -2}
\][/tex]
