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Answer :
To simplify the expression [tex]\(-4 x^2 (6 x - 5 x^2 - 5)\)[/tex], we will distribute [tex]\(-4 x^2\)[/tex] to each term inside the parentheses. Let's go through this step by step:
1. Distribute to the first term:
[tex]\[
-4 x^2 \cdot 6 x = -24 x^3
\][/tex]
Here, we multiply [tex]\(-4 x^2\)[/tex] by [tex]\(6 x\)[/tex]. The coefficients [tex]\(-4\)[/tex] and [tex]\(6\)[/tex] are multiplied to get [tex]\(-24\)[/tex], and [tex]\(x^2 \cdot x\)[/tex] gives [tex]\(x^{2+1} = x^3\)[/tex].
2. Distribute to the second term:
[tex]\[
-4 x^2 \cdot (-5 x^2) = 20 x^4
\][/tex]
Now, we multiply [tex]\(-4 x^2\)[/tex] by [tex]\(-5 x^2\)[/tex]. The coefficients [tex]\(-4\)[/tex] and [tex]\(-5\)[/tex] give [tex]\(20\)[/tex] when multiplied (negative times negative gives positive), and [tex]\(x^2 \cdot x^2\)[/tex] results in [tex]\(x^{2+2} = x^4\)[/tex].
3. Distribute to the third term:
[tex]\[
-4 x^2 \cdot (-5) = 20 x^2
\][/tex]
Finally, we multiply [tex]\(-4 x^2\)[/tex] by [tex]\(-5\)[/tex]. Here, the coefficients are [tex]\(-4\)[/tex] and [tex]\(-5\)[/tex], which multiply to [tex]\(20\)[/tex], and since there is no [tex]\(x\)[/tex] term to multiply with, the [tex]\(x^2\)[/tex] remains [tex]\(x^2\)[/tex].
Putting it all together, the simplified expression is:
[tex]\[
20 x^4 - 24 x^3 + 20 x^2
\][/tex]
Therefore, the correct simplification of the expression is:
[tex]\[ 20 x^4 - 24 x^3 + 20 x^2 \][/tex]
This matches the third option in the list.
1. Distribute to the first term:
[tex]\[
-4 x^2 \cdot 6 x = -24 x^3
\][/tex]
Here, we multiply [tex]\(-4 x^2\)[/tex] by [tex]\(6 x\)[/tex]. The coefficients [tex]\(-4\)[/tex] and [tex]\(6\)[/tex] are multiplied to get [tex]\(-24\)[/tex], and [tex]\(x^2 \cdot x\)[/tex] gives [tex]\(x^{2+1} = x^3\)[/tex].
2. Distribute to the second term:
[tex]\[
-4 x^2 \cdot (-5 x^2) = 20 x^4
\][/tex]
Now, we multiply [tex]\(-4 x^2\)[/tex] by [tex]\(-5 x^2\)[/tex]. The coefficients [tex]\(-4\)[/tex] and [tex]\(-5\)[/tex] give [tex]\(20\)[/tex] when multiplied (negative times negative gives positive), and [tex]\(x^2 \cdot x^2\)[/tex] results in [tex]\(x^{2+2} = x^4\)[/tex].
3. Distribute to the third term:
[tex]\[
-4 x^2 \cdot (-5) = 20 x^2
\][/tex]
Finally, we multiply [tex]\(-4 x^2\)[/tex] by [tex]\(-5\)[/tex]. Here, the coefficients are [tex]\(-4\)[/tex] and [tex]\(-5\)[/tex], which multiply to [tex]\(20\)[/tex], and since there is no [tex]\(x\)[/tex] term to multiply with, the [tex]\(x^2\)[/tex] remains [tex]\(x^2\)[/tex].
Putting it all together, the simplified expression is:
[tex]\[
20 x^4 - 24 x^3 + 20 x^2
\][/tex]
Therefore, the correct simplification of the expression is:
[tex]\[ 20 x^4 - 24 x^3 + 20 x^2 \][/tex]
This matches the third option in the list.
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