To solve the equation
[tex]\[
\left(\frac{-2}{5}\right)^{-3} \times \left(\frac{-2}{5}\right)^{11} = \left(\frac{-2}{5}\right)^{3x + 2},
\][/tex]
we can use the properties of exponents. Specifically, when multiplying two expressions with the same base, we add their exponents. Thus, the left-hand side of the equation can be simplified as follows:
[tex]\[
\left(\frac{-2}{5}\right)^{-3} \times \left(\frac{-2}{5}\right)^{11} = \left(\frac{-2}{5}\right)^{-3 + 11}.
\][/tex]
Next, we calculate the exponent on the left-hand side:
[tex]\[
-3 + 11 = 8.
\][/tex]
So, the equation now becomes:
[tex]\[
\left(\frac{-2}{5}\right)^{8} = \left(\frac{-2}{5}\right)^{3x + 2}.
\][/tex]
Since the bases are the same on both sides of the equation, we can set the exponents equal to each other:
[tex]\[
8 = 3x + 2.
\][/tex]
Now, solve for [tex]\(x\)[/tex]. First, subtract 2 from both sides of the equation:
[tex]\[
8 - 2 = 3x,
\][/tex]
which simplifies to:
[tex]\[
6 = 3x.
\][/tex]
Next, divide both sides by 3:
[tex]\[
x = \frac{6}{3}.
\][/tex]
Thus, we find:
[tex]\[
x = 2.
\][/tex]
Therefore, the value of [tex]\(x\)[/tex] that satisfies the equation is [tex]\(x = 2\)[/tex].
