1. **State the problem:** We are given four numbers: $-6$, $4$, $-1$, and $-2$. We need to find an equation involving these numbers using multiplication and addition. 2. **Understand the goal:** Typically, such problems ask to find a polynomial equation whose roots are these numbers. The polynomial with roots $r_1, r_2, r_3, r_4$ is given by: $$ (x - r_1)(x - r_2)(x - r_3)(x - r_4) = 0 $$ 3. **Apply the formula:** Here, the roots are $-6$, $4$, $-1$, and $-2$. So the polynomial is: $$ (x - (-6))(x - 4)(x - (-1))(x - (-2)) = 0 $$ which simplifies to: $$ (x + 6)(x - 4)(x + 1)(x + 2) = 0 $$ 4. **Multiply step-by-step:** First multiply the first two factors: $$ (x + 6)(x - 4) = x^2 - 4x + 6x - 24 = x^2 + 2x - 24 $$ Next multiply the last two factors: $$ (x + 1)(x + 2) = x^2 + 2x + x + 2 = x^2 + 3x + 2 $$ 5. **Multiply the two quadratics:** $$ (x^2 + 2x - 24)(x^2 + 3x + 2) $$ Multiply each term: $$ x^2 \cdot x^2 = x^4 $$ $$ x^2 \cdot 3x = 3x^3 $$ $$ x^2 \cdot 2 = 2x^2 $$ $$ 2x \cdot x^2 = 2x^3 $$ $$ 2x \cdot 3x = 6x^2 $$ $$ 2x \cdot 2 = 4x $$ $$ -24 \cdot x^2 = -24x^2 $$ $$ -24 \cdot 3x = -72x $$ $$ -24 \cdot 2 = -48 $$ 6. **Combine like terms:** $$ x^4 + (3x^3 + 2x^3) + (2x^2 + 6x^2 - 24x^2) + (4x - 72x) - 48 $$ $$ = x^4 + 5x^3 - 16x^2 - 68x - 48 $$ 7. **Final equation:** $$ x^4 + 5x^3 - 16x^2 - 68x - 48 = 0 $$ This is the polynomial equation with roots $-6$, $4$, $-1$, and $-2$ using multiplication and addition.