Step 1 :Set the two equations equal to each other to find the points of intersection: \(6x = x^2 - 7\)
Step 2 :Rearrange the equation to form a quadratic equation: \(x^2 - 6x - 7 = 0\)
Step 3 :Factor the quadratic equation to find the roots: \((x - 7)(x + 1) = 0\)
Step 4 :The points of intersection are \(x = -1\) and \(x = 7\)
Step 5 :We are only interested in the region between \(x = -5\) and \(x = 1\)
Step 6 :For \(x\) between \(-5\) and \(-1\), the curve \(y = 6x\) is above the curve \(y = x^2 - 7\)
Step 7 :For \(x\) between \(-1\) and \(1\), the curve \(y = x^2 - 7\) is above the curve \(y = 6x\)
Step 8 :The area between the curves is given by the integral \(\int_{-5}^{-1}[(6x) - (x^2 - 7)] dx + \int_{-1}^{1}[(x^2 - 7) - (6x)] dx\)
Step 9 :Simplify the integrals to get \(\int_{-5}^{-1}[-x^2 + 6x + 7] dx + \int_{-1}^{1}[x^2 - 6x + 7] dx\)
Step 10 :\(\boxed{\int_{-5}^{-1}[-x^2 + 6x + 7] dx + \int_{-1}^{1}[x^2 - 6x + 7] dx}\) is the final answer