vanessa uses the expressions $(3x^{2}+5x + 10)$ and $(x^{2}-3x - 1)$ to represent the length and width of…

vanessa uses the expressions $(3x^{2}+5x + 10)$ and $(x^{2}-3x - 1)$ to represent the length and width of her patio. which expression represents the area ($lw$) of vanessas patio?\n$3x^{4}-4x^{3}-8x^{2}-35x - 10$\n$3x^{4}+14x^{3}+22x^{2}+25x - 10$\n$3x^{4}-4x^{3}+22x^{2}+35x + 10$\n$3x^{4}+14x^{3}+28x^{2}+35x + 10$

vanessa uses the expressions $(3x^{2}+5x + 10)$ and $(x^{2}-3x - 1)$ to represent the length and width of her patio. which expression represents the area ($lw$) of vanessas patio?\n$3x^{4}-4x^{3}-8x^{2}-35x - 10$\n$3x^{4}+14x^{3}+22x^{2}+25x - 10$\n$3x^{4}-4x^{3}+22x^{2}+35x + 10$\n$3x^{4}+14x^{3}+28x^{2}+35x + 10$

Answer

Explanation:

Step1: Recall area formula

The area of a rectangle is $A = lw$. Here, $l=(3x^{2}+5x + 10)$ and $w=(x^{2}-3x - 1)$. So $A=(3x^{2}+5x + 10)(x^{2}-3x - 1)$.

Step2: Use distributive property

[ \begin{align*} &(3x^{2}+5x + 10)(x^{2}-3x - 1)\ =&3x^{2}(x^{2}-3x - 1)+5x(x^{2}-3x - 1)+10(x^{2}-3x - 1)\ =&(3x^{2}\cdot x^{2}-3x^{2}\cdot3x-3x^{2}\cdot1)+(5x\cdot x^{2}-5x\cdot3x - 5x\cdot1)+(10x^{2}-10\cdot3x-10\cdot1) \end{align*} ]

Step3: Simplify each term

[ \begin{align*} 3x^{2}\cdot x^{2}-3x^{2}\cdot3x-3x^{2}\cdot1&=3x^{4}-9x^{3}-3x^{2}\ 5x\cdot x^{2}-5x\cdot3x - 5x\cdot1&=5x^{3}-15x^{2}-5x\ 10x^{2}-10\cdot3x-10\cdot1&=10x^{2}-30x - 10 \end{align*} ]

Step4: Combine like - terms

[ \begin{align*} &(3x^{4}-9x^{3}-3x^{2})+(5x^{3}-15x^{2}-5x)+(10x^{2}-30x - 10)\ =&3x^{4}+(-9x^{3}+5x^{3})+(-3x^{2}-15x^{2}+10x^{2})+(-5x-30x)-10\ =&3x^{4}-4x^{3}-8x^{2}-35x - 10 \end{align*} ]

Answer:

$3x^{4}-4x^{3}-8x^{2}-35x - 10$