1. **State the problem:** Multiply the binomials $$\left(\frac{1}{4}x - 6\right)$$ and $$\left(\frac{4}{3}x - \frac{5}{4}\right)$$ and express the product as a trinomial in simplest form. 2. **Recall the formula:** To multiply two binomials $$(a + b)(c + d)$$, use the distributive property (FOIL method): $$ac + ad + bc + bd$$. 3. **Apply FOIL:** $$\left(\frac{1}{4}x\right)\left(\frac{4}{3}x\right) + \left(\frac{1}{4}x\right)\left(-\frac{5}{4}\right) + (-6)\left(\frac{4}{3}x\right) + (-6)\left(-\frac{5}{4}\right)$$ 4. **Multiply each term:** $$\frac{1}{4} \times \frac{4}{3} x \times x = \frac{1 \times 4}{4 \times 3} x^2 = \frac{4}{12} x^2$$ $$\frac{1}{4} x \times -\frac{5}{4} = -\frac{5}{16} x$$ $$-6 \times \frac{4}{3} x = -\frac{24}{3} x = -8 x$$ $$-6 \times -\frac{5}{4} = \frac{30}{4} = \frac{15}{2}$$ 5. **Simplify the first term:** $$\frac{4}{12} x^2 = \frac{1}{3} x^2$$ 6. **Combine like terms for $x$:** $$-\frac{5}{16} x - 8 x = -\frac{5}{16} x - \frac{128}{16} x = -\frac{133}{16} x$$ 7. **Write the final trinomial:** $$\frac{1}{3} x^2 - \frac{133}{16} x + \frac{15}{2}$$ This is the product expressed as a trinomial in simplest form.