Question

find the product.\\((2x^2 - 3x)(x^2 + 4x)\\)\\(\boldsymbol{?}x^4 + \boldsymbol{}x^3 + \boldsymbol{}x^2\\)

Explanation:

Step1: Apply distributive property (FOIL)

Multiply each term in the first polynomial by each term in the second polynomial:
$$(2x^2)(x^2) + (2x^2)(4x) + (-3x)(x^2) + (-3x)(4x)$$

Step2: Simplify each term

• For $(2x^2)(x^2)$: Using the rule $a^m \cdot a^n = a^{m + n}$, we get $2x^{2+2}=2x^4$.
• For $(2x^2)(4x)$: $2\times4\times x^{2 + 1}=8x^3$.
• For $(-3x)(x^2)$: $-3\times x^{1+2}=-3x^3$.
• For $(-3x)(4x)$: $-3\times4\times x^{1+1}=-12x^2$.

Step3: Combine like terms

Combine the $x^3$ terms: $8x^3-3x^3 = 5x^3$. The $x^4$ term is $2x^4$ and the $x^2$ term is $-12x^2$. So the polynomial is $2x^4+5x^3 - 12x^2$.

Answer:

For the $x^4$ term: $\boldsymbol{2}$; for the $x^3$ term: $\boldsymbol{5}$; for the $x^2$ term: $\boldsymbol{-12}$