Answer:
![16x^2+4x+1](https://img.qammunity.org/2023/formulas/mathematics/middle-school/4okllvtsfwqr42q6mcgkgs43x0ykqxc9jb.png)
Explanation:
Given expression:
![(64x^3-1)/(4x-1)](https://img.qammunity.org/2023/formulas/mathematics/middle-school/1gwndc1jvlt2t45vqyqmvhl1td0fieyn8z.png)
Step 1
Factor the numerator of the given expression.
Rewrite 64 as 4³ and 1 as 1³:
![\implies (4^3)x^3-1^3](https://img.qammunity.org/2023/formulas/mathematics/middle-school/28nn9lt5y4h1ayefogmydulc78nilz0sh8.png)
![\textsf{Apply the exponent rule} \quad a^b \cdot c^b=(ac)^b:](https://img.qammunity.org/2023/formulas/mathematics/middle-school/ebs9fg40l6xtvw270u7e7k0bjafvbkva5u.png)
![\implies (4x)^3-1^3](https://img.qammunity.org/2023/formulas/mathematics/middle-school/f836u5a7mvlb2nf074vq0b7q14k2f15k52.png)
![\textsf{Apply the Difference of Cubes Formula} \quad x^3-y^3=\left(x-y\right)\left(x^2+xy+y^2\right):](https://img.qammunity.org/2023/formulas/mathematics/middle-school/lhkoz3hoypapj8l1l3t8v0lm83myfkwxe8.png)
![\implies (4x)^3-1^3=(4x-1)\left((4x)^2+4x(1)+(1)^2\right)](https://img.qammunity.org/2023/formulas/mathematics/middle-school/ldd2a4g9p3shszpffc8i2mgifkwvyrcp65.png)
![\implies (4x)^3-1^3=(4x-1)\left(16x^2+4x+1\right)](https://img.qammunity.org/2023/formulas/mathematics/middle-school/drh6j2qg5och58hp1ksfie5b30o7vwbb06.png)
Step 2
Replace the numerator in the given expression with the factored numerator from step 1:
![\implies (64x^3-1)/(4x-1)=((4x-1)\left(16x^2+4x+1\right))/(4x-1)](https://img.qammunity.org/2023/formulas/mathematics/middle-school/42of3k0avradaldr2nmtelbcigxc4sghx7.png)
Cancel the common factor (4x - 1):
![\implies 16x^2+4x+1](https://img.qammunity.org/2023/formulas/mathematics/middle-school/dsvzpl8p4h2umri23ueqrk4lw4drqb095h.png)