1. The problem is to analyze the expression $$2x^2 + 5|x| - 4$$. 2. This expression involves a quadratic term $$2x^2$$, an absolute value term $$5|x|$$, and a constant $$-4$$. 3. To understand the behavior, consider two cases based on the absolute value: - Case 1: When $$x \geq 0$$, $$|x| = x$$, so the expression becomes $$2x^2 + 5x - 4$$. - Case 2: When $$x < 0$$, $$|x| = -x$$, so the expression becomes $$2x^2 - 5x - 4$$. 4. For each case, we can analyze or graph the quadratic function: - For $$x \geq 0$$: $$y = 2x^2 + 5x - 4$$. - For $$x < 0$$: $$y = 2x^2 - 5x - 4$$. 5. Both are quadratic functions opening upwards (since coefficient of $$x^2$$ is positive). 6. The expression is continuous and piecewise defined by these two quadratics. Final answer: The function is piecewise defined as $$y = \begin{cases} 2x^2 + 5x - 4 & x \geq 0 \\ 2x^2 - 5x - 4 & x < 0 \end{cases}$$.