1. **State the problem:** Simplify the expression $$(4m-1)^2 - 3m(y+x)^2$$. 2. **Recall formulas:** - Square of a binomial: $$(a-b)^2 = a^2 - 2ab + b^2$$ - Square of a sum: $$(y+x)^2 = y^2 + 2yx + x^2$$ 3. **Expand the first term:** $$ (4m-1)^2 = (4m)^2 - 2 \times 4m \times 1 + 1^2 = 16m^2 - 8m + 1 $$ 4. **Expand the second term:** $$ (y+x)^2 = y^2 + 2yx + x^2 $$ Multiply by $3m$: $$ 3m(y^2 + 2yx + x^2) = 3my^2 + 6myx + 3mx^2 $$ 5. **Combine the terms:** $$ 16m^2 - 8m + 1 - (3my^2 + 6myx + 3mx^2) $$ 6. **Write the final simplified expression:** $$ \boxed{16m^2 - 8m + 1 - 3my^2 - 6myx - 3mx^2} $$ This is the simplified form of the given expression.