1. **State the problem:** We are given three functions: - $a(x) = 4x^{2} + 2x - 5$ - $b(x) = \sqrt[3]{2x - 1}$ - $f(x) = \begin{cases} 2x + 5 & x \leq -3 \\ |x| & x > -3 \end{cases}$ We need to find and simplify $a(g)$, but the function $g$ is not explicitly given. Assuming $g = f$ or $g = b$, we clarify and proceed with $a(f(x))$ since $f$ is piecewise and more complex. 2. **Recall the composition of functions:** $$a(f(x)) = 4(f(x))^{2} + 2(f(x)) - 5$$ 3. **Evaluate $a(f(x))$ piecewise:** - For $x \leq -3$, $f(x) = 2x + 5$ $$a(f(x)) = 4(2x + 5)^{2} + 2(2x + 5) - 5$$ Expand $(2x + 5)^{2}$: $$ (2x + 5)^{2} = 4x^{2} + 20x + 25 $$ Substitute back: $$a(f(x)) = 4(4x^{2} + 20x + 25) + 4x + 10 - 5$$ Multiply out: $$= 16x^{2} + 80x + 100 + 4x + 10 - 5$$ Combine like terms: $$= 16x^{2} + 84x + 105$$ - For $x > -3$, $f(x) = |x|$ $$a(f(x)) = 4|x|^{2} + 2|x| - 5 = 4x^{2} + 2|x| - 5$$ 4. **Final simplified form:** $$a(f(x)) = \begin{cases} 16x^{2} + 84x + 105 & x \leq -3 \\ 4x^{2} + 2|x| - 5 & x > -3 \end{cases}$$ This is the simplified expression for $a(f(x))$.