To solve the integral

$\int (4x + 2) (x^2 + x)^7 \, dx,$

we can use substitution. Let

$u = x^2 + x.$

Then, the derivative $du$ is given by

$du = (2x + 1) \, dx.$

To express $dx$ in terms of $du$, we solve for $dx$:

$dx = \frac{du}{2x + 1}.$

Next, we need to express $4x + 2$ in terms of $u$. Notice:

$4x + 2 = 4(x^2 + x) + 2 - 4x^2 = 4u + 2 - 4x^2.$

However, since $x^2 = u - x$, this substitution becomes complex. Instead, let’s proceed by integrating by parts or using a simpler substitution.

Let's rewrite the integral as:

$\int (4x + 2) u^7 \, dx.$

Now, we can compute:

1. Substitute $u$ and $du$ back into the integral.
2. Simplify and integrate.

The new integral becomes:

$\int (4x + 2) u^7 \cdot \frac{du}{2x + 1}.$

However, the easier way is to notice that:

$\int (4x + 2) (x^2 + x)^7 \, dx = \int (4x + 2) \cdot u^7 \, dx,$

and integrate directly or recognize this form as a product of functions.

By applying integration by parts or recognizing the pattern of the product rule, we can solve this integral.

Would you like to explore a specific method or need a step-by-step guide on how to apply integration by parts here?