Let's analyze the given problem:

We are provided with the following definite integrals:

$\int_1^6 f(x) \, dx = -2, \quad \int_6^{11} f(x) \, dx = 17, \quad \int_1^6 g(x) \, dx = -1, \quad \int_6^{11} g(x) \, dx = 5$

The task is to calculate:

$\int_4^8 \left( f(x+3) - 2 \right) dx$

Step 1: Substitution for the function $f(x+3)$

Let’s perform a substitution in the integral. Set $u = x + 3$, which implies $du = dx$. The limits of integration will change accordingly:

• When $x = 4$, $u = 4 + 3 = 7$
• When $x = 8$, $u = 8 + 3 = 11$

Thus, the integral becomes: $\int_4^8 \left( f(x+3) - 2 \right) dx = \int_7^{11} f(u) \, du - \int_4^8 2 \, dx$

Step 2: Solve the transformed integral

Now, we need to calculate two parts:

1. First part: $\int_7^{11} f(u) \, du$ From the given data, we know: $\int_6^{11} f(x) \, dx = 17$ To get $\int_7^{11} f(u) \, du$, we use the property of definite integrals: $\int_6^{11} f(x) \, dx = \int_6^7 f(x) \, dx + \int_7^{11} f(x) \, dx$ Therefore: $17 = \int_6^7 f(x) \, dx + \int_7^{11} f(x) \, dx$ We also know from the first given integral: $\int_1^6 f(x) \, dx = -2$ Thus, $\int_6^7 f(x) \, dx$ can be found by subtracting $\int_1^6 f(x) \, dx$ from $\int_6^{11} f(x) \, dx$, but we lack this exact information about $\int_6^7 f(x) \, dx$.