Solve the following \quadratic equation by factoring. First, write the expression in completely factored form. Then write the real number solutions. -4x^{2} = 12x

Starting with the given equation: -4x^2 = 12x We can rearrange it to: -4x^2 - 12x = 0 And then factor out a common factor of -4x: -4x(x + 3) = 0 This gives us two possible solutions: -4x = 0, which simplifies to x = 0 Or: x + 3 = 0, which simplifies to x = -3 Therefore, the solutions to the \quadratic equation -4x^2 = 12x are x = 0 and x = -3. Explanation: To solve the \quadratic equation by factoring, we need to find two numbers that multiply together to give us -4 times 12, which is -48. Those two numbers must also add up to the coefficient of the x term, which is -12. We can see that -4 and 12 satisfy these conditions, so we factor out a common factor of -4x. This gives us: -4x(x + 3) = 0 Now we can use the zero product property, which states that if the product of two factors is equal to zero, then at least one of the factors must be zero. In this case, we have: -4x = 0 or x + 3 = 0 The first equation simplifies to x = 0, while the second equation simplifies to x = -3. Therefore, the solutions to the \quadratic equation are x = 0 and x = -3.