1. **Problem:** Determine if $f(a) = (2a^2 + 3a + 1)^2$ is equivalent to $g(a) = 4(a^4 + 1.5a^3 + 3a^2 + 2a + 0.25)$. Justify your answer using algebra. 2. **Formula and rules:** To check equivalence, expand both expressions fully and compare terms. 3. **Expand $f(a)$:** $$f(a) = (2a^2 + 3a + 1)^2 = (2a^2 + 3a + 1)(2a^2 + 3a + 1)$$ Multiply each term: $$= 2a^2 \times 2a^2 + 2a^2 \times 3a + 2a^2 \times 1 + 3a \times 2a^2 + 3a \times 3a + 3a \times 1 + 1 \times 2a^2 + 1 \times 3a + 1 \times 1$$ Calculate each: $$= 4a^4 + 6a^3 + 2a^2 + 6a^3 + 9a^2 + 3a + 2a^2 + 3a + 1$$ Combine like terms: $$= 4a^4 + (6a^3 + 6a^3) + (2a^2 + 9a^2 + 2a^2) + (3a + 3a) + 1$$ $$= 4a^4 + 12a^3 + 13a^2 + 6a + 1$$ 4. **Expand $g(a)$:** $$g(a) = 4(a^4 + 1.5a^3 + 3a^2 + 2a + 0.25)$$ Distribute 4: $$= 4a^4 + 6a^3 + 12a^2 + 8a + 1$$ 5. **Compare $f(a)$ and $g(a)$:** $$f(a) = 4a^4 + 12a^3 + 13a^2 + 6a + 1$$ $$g(a) = 4a^4 + 6a^3 + 12a^2 + 8a + 1$$ 6. **Conclusion:** Since the coefficients of $a^3$, $a^2$, and $a$ terms differ, $f(a) \neq g(a)$. **Final answer:** $f(a)$ is not equivalent to $g(a)$.