To solve the equation [tex]\(\left(\frac{1}{4}\right)^{x+1} = 32\)[/tex], let's go through the steps systematically:
1. Express 32 as a power of 4:
We know that [tex]\(32 = 2^5\)[/tex]. However, considering we want to express [tex]\(32\)[/tex] in terms of base [tex]\(4\)[/tex], we note that:
[tex]\[
32 = 4^{\frac{5}{2}}
\][/tex]
This is because [tex]\(4^{1.5} = 2^3\)[/tex], and further exponentiation of 2 leads to the power we need.
2. Rewrite the given equation:
The equation is:
[tex]\[
\left(\frac{1}{4}\right)^{x+1} = 32
\][/tex]
Notice that [tex]\(\frac{1}{4}\)[/tex] can be rewritten as [tex]\(4^{-1}\)[/tex]. Thus:
[tex]\[
\left(4^{-1}\right)^{x+1} = 32
\][/tex]
Simplify the left-hand side to get:
[tex]\[
4^{-(x+1)} = 32
\][/tex]
Using our earlier transformation for 32:
[tex]\[
4^{-(x+1)} = 4^{\frac{5}{2}}
\][/tex]
3. Set the exponents equal to each other:
Since we now have the same base, we can set the exponents equal:
[tex]\[
-(x+1) = \frac{5}{2}
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Simplify the equation:
[tex]\[
-(x+1) = \frac{5}{2}
\][/tex]
[tex]\[
x + 1 = -\frac{5}{2}
\][/tex]
Subtract 1 from both sides:
[tex]\[
x = -\frac{5}{2} - 1
\][/tex]
Convert 1 to a fraction format to combine:
[tex]\[
x = -\frac{5}{2} - \frac{2}{2}
\][/tex]
[tex]\[
x = -\frac{7}{2}
\][/tex]
Thus, the solution to the equation [tex]\(\left(\frac{1}{4}\right)^{x+1} = 32\)[/tex] is
[tex]\[
\boxed{-\frac{7}{2}}
\][/tex]
Therefore, the correct answer is [tex]\(A. -\frac{7}{2}\)[/tex].
