\( \int _ { - 6 } ^ { - 2 } ( x + 4 ) e ^ { x ^ { 2 } + 8 x + 12 } d x = \square \)

Calculate the integral \( \int_{-6}^{-2} (x+4)e^{x^2+8x+12}dx \). Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int_{-6}^{-2} \left(x+4\right)e^{x^{2}+8x+12} dx\) - step1: Evaluate the integral: \(\int \left(x+4\right)e^{x^{2}+8x+12} dx\) - step2: Use the substitution \(dx=\frac{1}{2x+8} dt\) to transform the integral\(:\) \(\int \left(x+4\right)e^{x^{2}+8x+12}\times \frac{1}{2x+8} dt\) - step3: Simplify: \(\int \frac{e^{x^{2}+8x+12}}{2} dt\) - step4: Use the substitution \(t=x^{2}+8x+12\) to transform the integral\(:\) \(\int \frac{e^{t}}{2} dt\) - step5: Rewrite the expression: \(\int \frac{1}{2}e^{t} dt\) - step6: Use properties of integrals: \(\frac{1}{2}\times \int e^{t} dt\) - step7: Evaluate the integral: \(\frac{1}{2}e^{t}\) - step8: Multiply the terms: \(\frac{e^{t}}{2}\) - step9: Substitute back: \(\frac{e^{x^{2}+8x+12}}{2}\) - step10: Return the limits: \(\left(\frac{e^{x^{2}+8x+12}}{2}\right)\bigg |_{-6}^{-2}\) - step11: Calculate the value: \(0\) The integral \( \int _ { - 6 } ^ { - 2 } ( x + 4 ) e ^ { x ^ { 2 } + 8 x + 12 } d x \) evaluates to 0.