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Answer :
To multiply
[tex]$$
\left(x^2-5x\right)\left(2x^2+x-3\right),
$$[/tex]
we apply the distributive property (also known as the FOIL method for binomials, extended to a binomial and a trinomial). This means each term in the first parentheses is multiplied by each term in the second parentheses.
1. Multiply the first term of the first factor, [tex]\(x^2\)[/tex], by each term in the second factor:
[tex]\[
x^2 \cdot 2x^2 = 2x^4,
\][/tex]
[tex]\[
x^2 \cdot x = x^3,
\][/tex]
[tex]\[
x^2 \cdot (-3) = -3x^2.
\][/tex]
2. Multiply the second term of the first factor, [tex]\(-5x\)[/tex], by each term in the second factor:
[tex]\[
-5x \cdot 2x^2 = -10x^3,
\][/tex]
[tex]\[
-5x \cdot x = -5x^2,
\][/tex]
[tex]\[
-5x \cdot (-3) = 15x.
\][/tex]
3. Now, list all the products:
[tex]\[
2x^4, \quad x^3, \quad -3x^2, \quad -10x^3, \quad -5x^2, \quad 15x.
\][/tex]
4. Combine like terms by grouping the terms with the same power of [tex]\(x\)[/tex]:
- The [tex]\(x^4\)[/tex] term is:
[tex]\[
2x^4.
\][/tex]
- The [tex]\(x^3\)[/tex] terms are:
[tex]\[
x^3 - 10x^3 = -9x^3.
\][/tex]
- The [tex]\(x^2\)[/tex] terms are:
[tex]\[
-3x^2 - 5x^2 = -8x^2.
\][/tex]
- The [tex]\(x\)[/tex] term is:
[tex]\[
15x.
\][/tex]
5. Therefore, the final expanded expression is:
[tex]$$
2x^4 - 9x^3 - 8x^2 + 15x.
$$[/tex]
Comparing with the options provided, the correct match is:
D. [tex]\(2x^4-9x^3-8x^2+15x\)[/tex].
[tex]$$
\left(x^2-5x\right)\left(2x^2+x-3\right),
$$[/tex]
we apply the distributive property (also known as the FOIL method for binomials, extended to a binomial and a trinomial). This means each term in the first parentheses is multiplied by each term in the second parentheses.
1. Multiply the first term of the first factor, [tex]\(x^2\)[/tex], by each term in the second factor:
[tex]\[
x^2 \cdot 2x^2 = 2x^4,
\][/tex]
[tex]\[
x^2 \cdot x = x^3,
\][/tex]
[tex]\[
x^2 \cdot (-3) = -3x^2.
\][/tex]
2. Multiply the second term of the first factor, [tex]\(-5x\)[/tex], by each term in the second factor:
[tex]\[
-5x \cdot 2x^2 = -10x^3,
\][/tex]
[tex]\[
-5x \cdot x = -5x^2,
\][/tex]
[tex]\[
-5x \cdot (-3) = 15x.
\][/tex]
3. Now, list all the products:
[tex]\[
2x^4, \quad x^3, \quad -3x^2, \quad -10x^3, \quad -5x^2, \quad 15x.
\][/tex]
4. Combine like terms by grouping the terms with the same power of [tex]\(x\)[/tex]:
- The [tex]\(x^4\)[/tex] term is:
[tex]\[
2x^4.
\][/tex]
- The [tex]\(x^3\)[/tex] terms are:
[tex]\[
x^3 - 10x^3 = -9x^3.
\][/tex]
- The [tex]\(x^2\)[/tex] terms are:
[tex]\[
-3x^2 - 5x^2 = -8x^2.
\][/tex]
- The [tex]\(x\)[/tex] term is:
[tex]\[
15x.
\][/tex]
5. Therefore, the final expanded expression is:
[tex]$$
2x^4 - 9x^3 - 8x^2 + 15x.
$$[/tex]
Comparing with the options provided, the correct match is:
D. [tex]\(2x^4-9x^3-8x^2+15x\)[/tex].
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