Question

using the distributive property, marta multiplied the binomial $(2x + 3)$ by the trinomial $(x^2 + x - 2)$ and got the given expression.
$(2x)(x^2) + (2x)(x) + (2x)(-2) + (3)(x^2) + (3)(x) + (3)(-2)$
which is the simplified product?
$2x^3 + x^2 - x - 6$
$2x^3 + 5x^2 - x - 6$
$2x^3 - x^2 - 7x - 6$
$2x^3 + 6x^2 - x - 6$

Explanation:

Step1: Simplify each term

• For \((2x)(x^2)\): Using the rule \(a^m\times a^n=a^{m + n}\), we have \(2x\times x^2 = 2x^{1+2}=2x^3\)
• For \((2x)(x)\): \(2x\times x=2x^{1 + 1}=2x^2\)
• For \((2x)(-2)\): \(2x\times(-2)=-4x\)
• For \((3)(x^2)\): \(3\times x^2 = 3x^2\)
• For \((3)(x)\): \(3\times x=3x\)
• For \((3)(-2)\): \(3\times(-2)=-6\)

Step2: Combine like terms

Now we have the terms: \(2x^3,2x^2,-4x,3x^2,3x,-6\)

• Combine the \(x^2\) terms: \(2x^2+3x^2 = 5x^2\)
• Combine the \(x\) terms: \(-4x + 3x=-x\)

Step3: Write the simplified polynomial

Putting all the simplified terms together: \(2x^3+5x^2 - x-6\)

Answer:

\(2x^3 + 5x^2 - x - 6\) (the second option)