1. **Stating the problem:** We want to simplify or understand the expression $$|x_{i-1} + 4|4|x_{i-1}|$$. 2. **Understanding the expression:** The expression contains an absolute value around $$x_{i-1} + 4$$ and then multiplies by 4 and by $$x_{i-1}$$ outside the absolute value. 3. **Recall properties of absolute values:** - $$|a| \geq 0$$ for any real number $$a$$. - $$|ab| = |a||b|$$. 4. **Rewrite the expression:** The expression is $$|x_{i-1} + 4| \times 4 \times |x_{i-1}|$$. 5. **Simplify step-by-step:** Since 4 is positive, $$|4| = 4$$, so we can write: $$|x_{i-1} + 4| \times 4 \times |x_{i-1}| = 4 \times |x_{i-1} + 4| \times |x_{i-1}|$$ 6. **Final simplified form:** $$4 |x_{i-1} + 4| |x_{i-1}|$$ This is the simplified expression showing the product of the absolute values and the constant 4. **Answer:** $$4 |x_{i-1} + 4| |x_{i-1}|$$