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Answer :
To multiply the polynomials
$$ (5x^2 - x - 3)(2x + 6), $$
we use the distributive property (also known as the FOIL method for binomials, extended here for a trinomial and a binomial).
1. Multiply each term in the first polynomial by each term in the second polynomial:
- Multiply $5x^2$ by each term of $2x + 6$:
$$ 5x^2 \cdot 2x = 10x^3, $$
$$ 5x^2 \cdot 6 = 30x^2. $$
- Multiply $-x$ by each term of $2x + 6$:
$$ -x \cdot 2x = -2x^2, $$
$$ -x \cdot 6 = -6x. $$
- Multiply $-3$ by each term of $2x + 6$:
$$ -3 \cdot 2x = -6x, $$
$$ -3 \cdot 6 = -18. $$
2. Now, combine the like terms:
- The cubic term:
$$ 10x^3. $$
- The quadratic terms:
$$ 30x^2 - 2x^2 = 28x^2. $$
- The linear terms:
$$ -6x - 6x = -12x. $$
- The constant term:
$$ -18. $$
3. Putting it all together, the product is:
$$ 10x^3 + 28x^2 - 12x - 18. $$
Thus, the final answer is:
$$ 10x^3 + 28x^2 - 12x - 18, $$
which corresponds to option A.
$$ (5x^2 - x - 3)(2x + 6), $$
we use the distributive property (also known as the FOIL method for binomials, extended here for a trinomial and a binomial).
1. Multiply each term in the first polynomial by each term in the second polynomial:
- Multiply $5x^2$ by each term of $2x + 6$:
$$ 5x^2 \cdot 2x = 10x^3, $$
$$ 5x^2 \cdot 6 = 30x^2. $$
- Multiply $-x$ by each term of $2x + 6$:
$$ -x \cdot 2x = -2x^2, $$
$$ -x \cdot 6 = -6x. $$
- Multiply $-3$ by each term of $2x + 6$:
$$ -3 \cdot 2x = -6x, $$
$$ -3 \cdot 6 = -18. $$
2. Now, combine the like terms:
- The cubic term:
$$ 10x^3. $$
- The quadratic terms:
$$ 30x^2 - 2x^2 = 28x^2. $$
- The linear terms:
$$ -6x - 6x = -12x. $$
- The constant term:
$$ -18. $$
3. Putting it all together, the product is:
$$ 10x^3 + 28x^2 - 12x - 18. $$
Thus, the final answer is:
$$ 10x^3 + 28x^2 - 12x - 18, $$
which corresponds to option A.
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