Let the number be denoted as $x$.

Step-by-step breakdown:

1. Expression for "four is added to six times a number": This part translates to: $6x + 4$

2. Squaring the result: Squaring the expression $6x + 4$, we get: $(6x + 4)^2$

3. Expression for "four times the square of the sum of the number and its next multiple": The next multiple of $x$ is $x + 1$. So, the sum of the number and its next multiple is: $x + (x + 1) = 2x + 1$ The square of this sum is $(2x + 1)^2$, and four times this square is: $4 \times (2x + 1)^2$

4. Equating the two expressions: The problem states that the square of $6x + 4$ is equal to four times the square of $2x + 1$: $(6x + 4)^2 = 4 \times (2x + 1)^2$

5. Solve the equation: Expand both sides: $(6x + 4)^2 = 36x^2 + 48x + 16$ $4 \times (2x + 1)^2 = 4 \times (4x^2 + 4x + 1) = 16x^2 + 16x + 4$

   Now, set the two expressions equal: $36x^2 + 48x + 16 = 16x^2 + 16x + 4$

6. Simplifying the equation: Subtract $16x^2 + 16x + 4$ from both sides: $20x^2 + 32x + 12 = 0$ Divide through by 4: $5x^2 + 8x + 3 = 0$

7. Solve the quadratic equation: Using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a = 5$, $b = 8$, and $c = 3$: $x = \frac{-8 \pm \sqrt{8^2 - 4(5)(3)}}{2(5)} = \frac{-8 \pm \sqrt{64 - 60}}{10} = \frac{-8 \pm \sqrt{4}}{10}$ $x = \frac{-8 \pm 2}{10}$

   So, $x = \frac{-8 + 2}{10} = \frac{-6}{10} = -\frac{3}{5}$, or $x = \frac{-8 - 2}{10} = \frac{-10}{10} = -1$.

8. Find twice the number: If $x = -1$, twice the number is: $2 \times (-1) = -2$

Thus, twice the number is -2.

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Related Questions:

1. How do you expand a binomial expression?
2. How is the quadratic formula derived?
3. What are some methods to solve quadratic equations other than the quadratic formula?
4. Can you explain how to complete the square?
5. What is the difference between a quadratic and linear equation?

Tip: Always check your solutions by plugging them back into the original equation to verify their correctness.