6. \( \left(\frac{a}{4}-4\right)\left(\frac{a^{2}}{16}+a+16\right) \)

To factor the expression \( \left(\frac{a}{4}-4\right)\left(\frac{a^{2}}{16}+a+16\right) \), we can first simplify our approach by rewriting the first part. The term \( \frac{a}{4} - 4 \) can be expressed as \( \frac{a - 16}{4} \). For the second term, we can observe that \( \frac{a^{2}}{16} + a + 16 \) is quadratic in form. To make factoring easier, let's first multiply it by \( 16 \) (the denominator of the leading term) to obtain: \[ a^{2} + 16a + 256 \] Now, we can factor this quadratic expression. The expression \( a^2 + 16a + 256 \) factors as \( (a+16)(a+16) \) or \( (a+16)^2 \). Putting it all together, we have: \[ \left(\frac{a - 16}{4}\right)\left((a + 16)^2\right) \] This can be simplified further as: \[ \frac{(a - 16)(a + 16)^2}{4} \] And that's the final factored form! As a recap, we ended up breaking it down into manageable parts to see the factoring opportunities that lay within the expression.