1. **State the problem:** Simplify the expression $$(3x+4) \times (3x + 4p) \times (7x - 1)$$. 2. **Multiply the first two binomials:** Use the distributive property (FOIL) for $$(3x+4)(3x+4p)$$. $$ (3x+4)(3x+4p) = 3x \times 3x + 3x \times 4p + 4 \times 3x + 4 \times 4p $$ $$ = 9x^2 + 12xp + 12x + 16p $$ 3. **Rewrite the expression:** Now the expression is $$ (9x^2 + 12xp + 12x + 16p)(7x - 1) $$ 4. **Multiply the result by the third binomial:** Distribute each term in the first polynomial by each term in the second binomial. $$ 9x^2 \times 7x = 63x^3 $$ $$ 9x^2 \times (-1) = -9x^2 $$ $$ 12xp \times 7x = 84x^2p $$ $$ 12xp \times (-1) = -12xp $$ $$ 12x \times 7x = 84x^2 $$ $$ 12x \times (-1) = -12x $$ $$ 16p \times 7x = 112xp $$ $$ 16p \times (-1) = -16p $$ 5. **Combine all terms:** $$ 63x^3 - 9x^2 + 84x^2p - 12xp + 84x^2 - 12x + 112xp - 16p $$ 6. **Group like terms:** $$ 63x^3 + (-9x^2 + 84x^2p + 84x^2) + (-12xp + 112xp) + (-12x) - 16p $$ 7. **Simplify each group:** $$ -9x^2 + 84x^2 = 75x^2 $$ $$ -12xp + 112xp = 100xp $$ 8. **Final simplified expression:** $$ 63x^3 + 75x^2 + 84x^2p + 100xp - 12x - 16p $$ This is the fully expanded and simplified form of the original expression.