Suppose $f^{\prime}(x)=8 x^{3}+12 x+2$ and $f(1)=-4$. Then $f(-1)$ equals (Enter a number for your answer.)

Step 1 ：Given that \(f'(x) = 8x^3 + 12x + 2\), we find the antiderivative \(f(x)\) by integrating \(f'(x)\) with respect to \(x\).

Step 2 ：The integral of \(8x^3\) is \(2x^4\), the integral of \(12x\) is \(6x^2\), and the integral of \(2\) is \(2x\). Therefore, the antiderivative of \(f'(x)\) is \(f(x) = 2x^4 + 6x^2 + 2x + C\), where \(C\) is the constant of integration.

Step 3 ：We know that \(f(1) = -4\), so we substitute \(x = 1\) into \(f(x)\) to solve for \(C\): \(-4 = 2(1)^4 + 6(1)^2 + 2(1) + C = 2 + 6 + 2 + C = 10 + C\).

Step 4 ：Solving for \(C\) gives \(C = -4 - 10 = -14\).

Step 5 ：So, \(f(x) = 2x^4 + 6x^2 + 2x - 14\).

Step 6 ：Finally, we substitute \(x = -1\) into \(f(x)\) to find \(f(-1)\): \(f(-1) = 2(-1)^4 + 6(-1)^2 + 2(-1) - 14 = 2 + 6 - 2 - 14 = -8\).

Step 7 ：\(\boxed{f(-1) = -8}\)