We are tasked with solving the integral:

\[
\int (t - 2) \sqrt{1 + t} \, dt
\]

### Step 1: Use substitution
We can simplify the integral by using a substitution. Let:

\[
u = 1 + t
\]

Then, the differential becomes:

\[
du = dt
\]
\[
t = u - 1
\]

Substitute these into the original integral:

\[
\int (t - 2) \sqrt{1 + t} \, dt = \int ((u - 1) - 2) \sqrt{u} \, du
\]

Simplifying the expression inside the integral:

\[
(u - 1 - 2) = u - 3
\]

So, the integral becomes:

\[
\int (u - 3) \sqrt{u} \, du
\]

### Step 2: Expand and simplify
Now, distribute \( \sqrt{u} \) across the terms in the parentheses:

\[
\int (u - 3) \sqrt{u} \, du = \int u^{3/2} \, du - 3 \int u^{1/2} \, du
\]

### Step 3: Integrate each term
Now, we can integrate each term separately.

1. For \( \int u^{3/2} \, du \), use the power rule for integration:
   \[
   \int u^{3/2} \, du = \frac{u^{5/2}}{\frac{5}{2}} = \frac{2}{5} u^{5/2}
   \]

2. For \( \int u^{1/2} \, du \), again use the power rule:
   \[
   \int u^{1/2} \, du = \frac{u^{3/2}}{\frac{3}{2}} = \frac{2}{3} u^{3/2}
   \]

Now, substitute these into the integral:

\[
\frac{2}{5} u^{5/2} - 3 \times \frac{2}{3} u^{3/2}
\]

Simplifying:

\[
\frac{2}{5} u^{5/2} - 2 u^{3/2}
\]

### Step 4: Substitute back \( u = 1 + t \)
Now, substitute back \( u = 1 + t \):

\[
\frac{2}{5} (1 + t)^{5/2} - 2 (1 + t)^{3/2} + C
\]

Thus, the final answer is:

\[
\boxed{\frac{2}{5} (1 + t)^{5/2} - 2 (1 + t)^{3/2} + C}
\]

Let me know if you want details or have any questions!  
Here are some related questions to explore further:

1. How can we verify this integral using differentiation?
2. What happens if we change the power of the square root to another value?
3. How would this integral change if the term \( (t - 2) \) were replaced by a different linear expression?
4. Can this method of substitution be applied to other similar integrals?
5. How do we deal with integrals involving cube roots instead of square roots?

**Tip**: When integrating functions involving square roots, substitution is a powerful tool for simplifying expressions.

Step-by-step guide to solving the integral \( \int (t - 2) \sqrt{1 + t} \, dt \) using substitution. This problem covers key concepts in integration and algebraic manipulation, suitable for students in Grades 11-12 and early college. Learn the application of substitution and power rule for solving complex integrals.

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