Certainly! Let's tackle the given problem step-by-step.
We are given the equation:
[tex]\[
(3x + 2)^2 + 7(3x + 2) - 8 = 0
\][/tex]
We will use substitution to simplify this equation. Let's choose a substitution to make the equation easier to handle.
Define [tex]\( u \)[/tex] as:
[tex]\[
u = 3x + 2
\][/tex]
Now, substitute [tex]\( u \)[/tex] back into the equation. The given equation simplifies to:
[tex]\[
u^2 + 7u - 8 = 0
\][/tex]
Hence, the equivalent quadratic equation in terms of [tex]\( u \)[/tex] is:
[tex]\[
u^2 + 7u - 8 = 0
\][/tex]
Remember that [tex]\( u = 3x + 2 \)[/tex]. Therefore, we have successfully written the equivalent quadratic equation as:
[tex]\[
u^2 + 7u - 8 = 0, \quad \text{where} \quad u = 3x + 2
\][/tex]
To summarize, the appropriate substitution results in:
[tex]\[
\boxed{u^2 + 7u - 8 = 0, \quad \text{where} \quad u = 3x + 2}
\][/tex]
This correctly captures the transformed form of the original equation.
