The given mathematical expression is:

$(6m^2 - 8mn + 4n^2)(8m + 8n)$

Solution:

We will expand this expression using the distributive property, multiplying each term of the first polynomial by each term of the second polynomial.

---

Step 1: Expand the expression

$(6m^2 - 8mn + 4n^2)(8m + 8n)$

Distribute $8m$ and $8n$ across each term of $6m^2 - 8mn + 4n^2$: $= 8m(6m^2 - 8mn + 4n^2) + 8n(6m^2 - 8mn + 4n^2)$

---

Step 2: Multiply $8m$ with each term in $6m^2 - 8mn + 4n^2$:

$8m(6m^2) = 48m^3$ $8m(-8mn) = -64m^2n$ $8m(4n^2) = 32mn^2$

Result: $8m(6m^2 - 8mn + 4n^2) = 48m^3 - 64m^2n + 32mn^2$

---

Step 3: Multiply $8n$ with each term in $6m^2 - 8mn + 4n^2$:

$8n(6m^2) = 48m^2n$ $8n(-8mn) = -64mn^2$ $8n(4n^2) = 32n^3$

Result: $8n(6m^2 - 8mn + 4n^2) = 48m^2n - 64mn^2 + 32n^3$

---

Step 4: Combine all terms:

$(6m^2 - 8mn + 4n^2)(8m + 8n) = (48m^3 - 64m^2n + 32mn^2) + (48m^2n - 64mn^2 + 32n^3)$

Group like terms: $= 48m^3 + (-64m^2n + 48m^2n) + (32mn^2 - 64mn^2) + 32n^3$

Simplify: $= 48m^3 - 16m^2n - 32mn^2 + 32n^3$

---

Final Answer:

$48m^3 - 16m^2n - 32mn^2 + 32n^3$

Would you like further clarification or breakdown of any step?

---

Related Questions:

1. How does the distributive property work in polynomial multiplication?
2. What are the steps to simplify like terms in a polynomial?
3. How can this polynomial be factored back?
4. What is the degree of the resulting polynomial?
5. Can you graph this polynomial function with specific values of $m$ and $n$?

Tip:

Always align terms by their degrees when simplifying polynomials to avoid errors.