Question

question 5.
the expression $3x^2 + 26x - 40$ is equivalent to the product
( )( )
(select)
$3x - 10$
$3x - 4$
$3x - 8$
$x + 5$

Explanation:

Step1: Test first factor option

Multiply $(3x-10)(x+5)$:

$$\begin{align*} (3x-10)(x+5)&=3x\cdot x + 3x\cdot5 -10\cdot x -10\cdot5\\ &=3x^2+15x-10x-50\\ &=3x^2+5x-50 \end{align*}$$

Step2: Test second factor option

Multiply $(3x-4)(x+5)$:

$$\begin{align*} (3x-4)(x+5)&=3x\cdot x + 3x\cdot5 -4\cdot x -4\cdot5\\ &=3x^2+15x-4x-20\\ &=3x^2+11x-20 \end{align*}$$

Step3: Test third factor option

Multiply $(3x-8)(x+5)$:

$$\begin{align*} (3x-8)(x+5)&=3x\cdot x + 3x\cdot5 -8\cdot x -8\cdot5\\ &=3x^2+15x-8x-40\\ &=3x^2+7x-40 \end{align*}$$

Step4: Correct factoring approach

Find two numbers $a,b$: $a\cdot b=3\cdot(-40)=-120$, $a+b=26$. The numbers are $30$ and $-4$.
Rewrite and factor:

$$\begin{align*} 3x^2+26x-40&=3x^2+30x-4x-40\\ &=3x(x+10)-4(x+10)\\ &=(3x-4)(x+10) \end{align*}$$

(Note: The given right factor $x+5$ is incorrect; the correct pair is $(3x-4)$ and $(x+10)$)

Answer:

The correct first factor is $\boldsymbol{3x - 4}$, and the correct second factor (replacing the incorrect $x+5$) is $\boldsymbol{x + 10}$. The product $(3x - 4)(x + 10)$ equals $3x^2 + 26x - 40$.