Answer :
We want to simplify the expression
$$
i^2(5x-2i)^2,
$$
where we know that the imaginary unit $i$ satisfies
$$
i^2 = -1.
$$
Let’s go through the steps:
1. First, expand the square:
$$
(5x-2i)^2 = (5x)^2 - 2\,(5x)(2i) + (2i)^2.
$$
Compute each term:
- $(5x)^2 = 25x^2.$
- $-2\,(5x)(2i) = -20xi.$
- $(2i)^2 = 4i^2.$
So the expansion gives:
$$
(5x-2i)^2 = 25x^2 - 20xi + 4i^2.
$$
2. Next, substitute $i^2 = -1$ into the expression:
$$
4i^2 = 4(-1) = -4.
$$
Thus, we have:
$$
(5x-2i)^2 = 25x^2 - 20xi - 4.
$$
3. Now, multiply this result by $i^2$ (which is $-1$):
$$
i^2(5x-2i)^2 = -1 \cdot (25x^2 - 20xi - 4) = -25x^2 + 20xi + 4.
$$
So the simplified expression is
$$
-25x^2 + 20xi + 4.
$$
Now, let’s compare this result with the given options:
1) $-25x^2 + 20xi - 4$
2) $-25x^2 + 20x + 4$
3) $25x^2 + 20xi + 4$
4) $25x^3 + 4$
We see that none of these match exactly with our expression $-25x^2+20xi+4$. The first option has the wrong constant term (it has $-4$ instead of $+4$), the second option lacks the imaginary unit $i$ in the $20x$ term, and the third and fourth options have other discrepancies.
Therefore, none of the provided options is correct. The final answer is:
$$
0 \quad \text{(none of the options)}
$$
$$
i^2(5x-2i)^2,
$$
where we know that the imaginary unit $i$ satisfies
$$
i^2 = -1.
$$
Let’s go through the steps:
1. First, expand the square:
$$
(5x-2i)^2 = (5x)^2 - 2\,(5x)(2i) + (2i)^2.
$$
Compute each term:
- $(5x)^2 = 25x^2.$
- $-2\,(5x)(2i) = -20xi.$
- $(2i)^2 = 4i^2.$
So the expansion gives:
$$
(5x-2i)^2 = 25x^2 - 20xi + 4i^2.
$$
2. Next, substitute $i^2 = -1$ into the expression:
$$
4i^2 = 4(-1) = -4.
$$
Thus, we have:
$$
(5x-2i)^2 = 25x^2 - 20xi - 4.
$$
3. Now, multiply this result by $i^2$ (which is $-1$):
$$
i^2(5x-2i)^2 = -1 \cdot (25x^2 - 20xi - 4) = -25x^2 + 20xi + 4.
$$
So the simplified expression is
$$
-25x^2 + 20xi + 4.
$$
Now, let’s compare this result with the given options:
1) $-25x^2 + 20xi - 4$
2) $-25x^2 + 20x + 4$
3) $25x^2 + 20xi + 4$
4) $25x^3 + 4$
We see that none of these match exactly with our expression $-25x^2+20xi+4$. The first option has the wrong constant term (it has $-4$ instead of $+4$), the second option lacks the imaginary unit $i$ in the $20x$ term, and the third and fourth options have other discrepancies.
Therefore, none of the provided options is correct. The final answer is:
$$
0 \quad \text{(none of the options)}
$$
