(4) $$ -2\left(-4 a^{2}+b^{3}\right)^{2} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
-2\left(-4 a^{2} + b^{3}\right)^{2}
$$

**Step 1: Expand the squared binomial**

Use the formula $$(x + y)^2 = x^2 + 2xy + y^2$$.

Let $$ x = -4a^{2} $$ and $$ y = b^{3} $$.

$$
\left(-4a^{2} + b^{3}\right)^{2} = (-4a^{2})^{2} + 2(-4a^{2})(b^{3}) + (b^{3})^{2}
$$

Calculate each term:

1. $$(-4a^{2})^{2} = 16a^{4}$$
2. $$2(-4a^{2})(b^{3}) = -8a^{2}b^{3}$$
3. $$(b^{3})^{2} = b^{6}$$

So,

$$
\left(-4a^{2} + b^{3}\right)^{2} = 16a^{4} - 8a^{2}b^{3} + b^{6}
$$

**Step 2: Multiply by -2**

$$
-2\left(16a^{4} - 8a^{2}b^{3} + b^{6}\right) = -32a^{4} + 16a^{2}b^{3} - 2b^{6}
$$

**Final Simplified Expression:**

$$
-32a^{4} + 16a^{2}b^{3} - 2b^{6}
$$
$$ -32a^{4} + 16a^{2}b^{3} - 2b^{6} $$

(4) \( -2\left(-4 a^{2}+b^{3}\right)^{2} \)