Introduction:
In this problem, we are given the differential equation dy/dt = 2t + 6 and the initial condition y(0) = 10. We are required to solve this equation and explain the steps in detail.

Solution:

Step 1: Separating Variables:
To solve the given differential equation, we need to separate the variables. We can write the equation as dy = (2t + 6) dt.

Step 2: Integrating Both Sides:
Next, we integrate both sides of the equation. The integral of dy is simply y, and the integral of (2t + 6) dt can be found by applying the power rule of integration.

∫dy = ∫(2t + 6) dt

y = t^2 + 6t + C

Here, C is the constant of integration.

Step 3: Applying Initial Condition:
Now, we can use the initial condition y(0) = 10 to find the value of the constant C. Substituting t = 0 and y = 10 into the equation, we get:

10 = 0^2 + 6(0) + C

10 = C

Therefore, the value of the constant C is 10.

Step 4: Final Solution:
Now that we have the value of the constant C, we can substitute it back into the equation to obtain the final solution.

y = t^2 + 6t + 10

This is the solution to the given differential equation with the initial condition.

Conclusion:
In conclusion, we have solved the differential equation dy/dt = 2t + 6 with the initial condition y(0) = 10. The solution is given by y = t^2 + 6t + 10.