\( f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 2 ; x = - 4 \)

To evaluate the function \( f(x) = 2x^3 + 3x^2 - 2 \) at \( x = -4 \), we substitute \(-4\) into the equation: \[ f(-4) = 2(-4)^3 + 3(-4)^2 - 2 \] Calculating step-by-step: - \((-4)^3 = -64\) so \(2(-64) = -128\), - \((-4)^2 = 16\) so \(3(16) = 48\), - Now putting it all together: \(-128 + 48 - 2 = -128 + 46 = -82\). Thus, \( f(-4) = -82 \). To check your work, breaking down such problems can help you avoid simple arithmetic errors. Double-check each power and coefficient as you go along. Also, when substituting negative numbers, it's easy to miss sign changes. Write out each step slowly to catch any mistakes!