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Answer :
To find the product of
[tex]$$
(2x^2 + 3x - 1) \quad \text{and} \quad (3x + 5),
$$[/tex]
we multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.
Step 1: Multiply the terms
1. Multiply [tex]$2x^2$[/tex] by each term in [tex]$(3x + 5)$[/tex]:
[tex]$$
2x^2 \cdot 3x = 6x^3,
$$[/tex]
[tex]$$
2x^2 \cdot 5 = 10x^2.
$$[/tex]
2. Multiply [tex]$3x$[/tex] by each term in [tex]$(3x + 5)$[/tex]:
[tex]$$
3x \cdot 3x = 9x^2,
$$[/tex]
[tex]$$
3x \cdot 5 = 15x.
$$[/tex]
3. Multiply [tex]$-1$[/tex] by each term in [tex]$(3x + 5)$[/tex]:
[tex]$$
-1 \cdot 3x = -3x,
$$[/tex]
[tex]$$
-1 \cdot 5 = -5.
$$[/tex]
Step 2: Write the expression with all terms
Combining all the terms together, we have:
[tex]$$
6x^3 + 10x^2 + 9x^2 + 15x - 3x - 5.
$$[/tex]
Step 3: Combine like terms
1. There is only one [tex]$x^3$[/tex] term:
[tex]$$
6x^3.
$$[/tex]
2. Combine the [tex]$x^2$[/tex] terms:
[tex]$$
10x^2 + 9x^2 = 19x^2.
$$[/tex]
3. Combine the [tex]$x$[/tex] terms:
[tex]$$
15x - 3x = 12x.
$$[/tex]
4. The constant term remains:
[tex]$$
-5.
$$[/tex]
Thus, the product is:
[tex]$$
6x^3 + 19x^2 + 12x - 5.
$$[/tex]
Final Answer: Option C.
[tex]$$
(2x^2 + 3x - 1) \quad \text{and} \quad (3x + 5),
$$[/tex]
we multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.
Step 1: Multiply the terms
1. Multiply [tex]$2x^2$[/tex] by each term in [tex]$(3x + 5)$[/tex]:
[tex]$$
2x^2 \cdot 3x = 6x^3,
$$[/tex]
[tex]$$
2x^2 \cdot 5 = 10x^2.
$$[/tex]
2. Multiply [tex]$3x$[/tex] by each term in [tex]$(3x + 5)$[/tex]:
[tex]$$
3x \cdot 3x = 9x^2,
$$[/tex]
[tex]$$
3x \cdot 5 = 15x.
$$[/tex]
3. Multiply [tex]$-1$[/tex] by each term in [tex]$(3x + 5)$[/tex]:
[tex]$$
-1 \cdot 3x = -3x,
$$[/tex]
[tex]$$
-1 \cdot 5 = -5.
$$[/tex]
Step 2: Write the expression with all terms
Combining all the terms together, we have:
[tex]$$
6x^3 + 10x^2 + 9x^2 + 15x - 3x - 5.
$$[/tex]
Step 3: Combine like terms
1. There is only one [tex]$x^3$[/tex] term:
[tex]$$
6x^3.
$$[/tex]
2. Combine the [tex]$x^2$[/tex] terms:
[tex]$$
10x^2 + 9x^2 = 19x^2.
$$[/tex]
3. Combine the [tex]$x$[/tex] terms:
[tex]$$
15x - 3x = 12x.
$$[/tex]
4. The constant term remains:
[tex]$$
-5.
$$[/tex]
Thus, the product is:
[tex]$$
6x^3 + 19x^2 + 12x - 5.
$$[/tex]
Final Answer: Option C.
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