\( \int ( 3 x + 2 ) ( 3 x ^ { 2 } + 4 x ) ^ { 4 } d x , \quad u = 3 x ^ { 2 } + 4 x \)

To solve the integral \( \int ( 3 x + 2 ) ( 3 x ^ { 2 } + 4 x ) ^ { 4 } d x \) using the substitution \( u = 3 x^{2} + 4 x \), we start by finding \( du \). The derivative of \( u \) with respect to \( x \) is: \[ du = (6x + 4) dx \] Now, we can express \( dx \) in terms of \( du \): \[ dx = \frac{du}{6x + 4} \] Next, we notice that \( 3x + 2 \) can be manipulated to relate to \( du \): \[ 3x + 2 = \frac{3}{2}(6x + 4) - 1 \] However, we can factor the expression for clarity. Now we can rearrange terms from the original triangle. Observing that \( (3x + 2) \) can be expressed through the modifier of \( du \), thus making the integral easier. Next, substituting \( u \) into the integral not only simplifies our expression significantly, but we reduce our need for terms involving \( x \) entirely thereby ensuring we’re aptly set to tackle \( \int u^{4} \) seamlessly afterwards. This integral thus transforms into: \[ \int \frac{(3x + 2)}{(6x + 4)} u^4 du \] This advanced manipulation will lead to a much clearer path towards solving the integral entirely in terms of \( u \), resolving any common mistakes related to the differentiation and substitution processes without getting tangled in \( x \) literals post-substitution. Finally, once integrated, we can revert back to the original variable by substituting \( u \) with \( 3x^2 + 4x \), thereby yielding an explicit solution grounded in our original context. Solution freshness is key here as we move forward, undaunted!