Divide \( a^{3}+64 b^{3} \) by \( a+4 b \) using long division

\[ \textbf{Step 1: Write the Dividend in Standard Form} \] Write the dividend \( a^3 + 64b^3 \) as: \[ a^3 + 0a^2 + 0a + 64b^3 \] \[ \textbf{Step 2: Divide the Leading Term} \] Divide the leading term of the dividend, \( a^3 \), by the leading term of the divisor, \( a \): \[ \frac{a^3}{a} = a^2 \] This gives the first term of the quotient, \( a^2 \). \[ \textbf{Step 3: Multiply and Subtract} \] Multiply the divisor \( a+4b \) by \( a^2 \): \[ a^2 \cdot (a+4b) = a^3 + 4ba^2 \] Subtract this from the dividend: \[ \begin{array}{rcl} \cline{1-1} \quad & a^3 + 0a^2 + 0a + 64b^3 \\ -(\, & a^3 + 4ba^2 & ) \\ \cline{1-1} \quad & 0a^3 - 4ba^2 + 0a + 64b^3 \\ \end{array} \] The new remainder is: \[ -4ba^2 + 0a + 64b^3 \] \[ \textbf{Step 4: Repeat the Process} \] Divide the new leading term \(-4ba^2\) by \( a \): \[ \frac{-4ba^2}{a} = -4ab \] This is the next term of the quotient. Multiply the divisor by \(-4ab\): \[ -4ab \cdot (a+4b) = -4ba^2 - 16ab^2 \] Subtract this product from the current remainder: \[ \begin{array}{rcl} \cline{1-1} \quad & -4ba^2 + 0a + 64b^3 \\ -(\, & -4ba^2 - 16ab^2 & ) \\ \cline{1-1} \quad & 0a^2 + 16ab^2 + 64b^3 \\ \end{array} \] The new remainder now is: \[ 16ab^2 + 64b^3 \] \[ \textbf{Step 5: Continue the Division} \] Divide the leading term \( 16ab^2 \) by \( a \): \[ \frac{16ab^2}{a} = 16b^2 \] This is the next term of the quotient. Multiply the divisor by \( 16b^2 \): \[ 16b^2 \cdot (a+4b) = 16ab^2 + 64b^3 \] Subtract this product from the current remainder: \[ \begin{array}{rcl} \cline{1-1} \quad & 16ab^2 + 64b^3 \\ -(\, & 16ab^2 + 64b^3 & ) \\ \cline{1-1} \quad & 0 \\ \end{array} \] The remainder is \( 0 \). \[ \textbf{Step 6: State the Final Quotient} \] The quotient obtained is: \[ a^2 - 4ab + 16b^2 \] Thus, dividing \( a^3+64b^3 \) by \( a+4b \) gives the quotient: \[ \boxed{a^2 - 4ab + 16b^2} \]