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Answer :
We want to multiply the polynomials
[tex]$$\left(5x^2 - x - 3\right)(2x + 6).$$[/tex]
Let’s compute it step by step.
1. Multiply every term in the first polynomial by every term in the second polynomial:
- Multiply the first term:
[tex]$$5x^2 \cdot (2x + 6) = 10x^3 + 30x^2.$$[/tex]
- Multiply the second term:
[tex]$$-x \cdot (2x + 6) = -2x^2 - 6x.$$[/tex]
- Multiply the third term:
[tex]$$-3 \cdot (2x + 6) = -6x - 18.$$[/tex]
2. Now, combine all the products:
[tex]$$10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18.$$[/tex]
3. Combine like terms:
- For [tex]$x^3$[/tex]: There is only [tex]$10x^3$[/tex].
- For [tex]$x^2$[/tex]: [tex]$30x^2 - 2x^2 = 28x^2.$[/tex]
- For [tex]$x$[/tex]: [tex]$-6x - 6x = -12x.$[/tex]
- The constant term is [tex]$-18$[/tex].
So the final product is:
[tex]$$10x^3 + 28x^2 - 12x - 18.$$[/tex]
Thus, the correct choice is:
Option C: [tex]$$10x^3 + 28x^2 - 12x - 18.$$[/tex]
[tex]$$\left(5x^2 - x - 3\right)(2x + 6).$$[/tex]
Let’s compute it step by step.
1. Multiply every term in the first polynomial by every term in the second polynomial:
- Multiply the first term:
[tex]$$5x^2 \cdot (2x + 6) = 10x^3 + 30x^2.$$[/tex]
- Multiply the second term:
[tex]$$-x \cdot (2x + 6) = -2x^2 - 6x.$$[/tex]
- Multiply the third term:
[tex]$$-3 \cdot (2x + 6) = -6x - 18.$$[/tex]
2. Now, combine all the products:
[tex]$$10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18.$$[/tex]
3. Combine like terms:
- For [tex]$x^3$[/tex]: There is only [tex]$10x^3$[/tex].
- For [tex]$x^2$[/tex]: [tex]$30x^2 - 2x^2 = 28x^2.$[/tex]
- For [tex]$x$[/tex]: [tex]$-6x - 6x = -12x.$[/tex]
- The constant term is [tex]$-18$[/tex].
So the final product is:
[tex]$$10x^3 + 28x^2 - 12x - 18.$$[/tex]
Thus, the correct choice is:
Option C: [tex]$$10x^3 + 28x^2 - 12x - 18.$$[/tex]
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