Solve by Factoring (5x+4)(x+1)=2

$(5 x + 4) (x + 1) = 2$

Step 1

Subtract $2$ from both sides of the equation.

$(5 x + 4) (x + 1) - 2 = 0$

Step 2

Simplify $(5 x + 4) (x + 1) - 2$ .

Simplify each term.

Expand $(5 x + 4) (x + 1)$ using the FOIL Method.

Apply the distributive property.

$5 x (x + 1) + 4 (x + 1) - 2 = 0$

Apply the distributive property.

$5 x \cdot x + 5 x \cdot 1 + 4 (x + 1) - 2 = 0$

Apply the distributive property.

$5 x \cdot x + 5 x \cdot 1 + 4 x + 4 \cdot 1 - 2 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$5 (x \cdot x) + 5 x \cdot 1 + 4 x + 4 \cdot 1 - 2 = 0$

Multiply $x$ by $x$ .

$5 x^{2} + 5 x \cdot 1 + 4 x + 4 \cdot 1 - 2 = 0$

Multiply $5$ by $1$ .

$5 x^{2} + 5 x + 4 x + 4 \cdot 1 - 2 = 0$

Multiply $4$ by $1$ .

$5 x^{2} + 5 x + 4 x + 4 - 2 = 0$

Add $5 x$ and $4 x$ .

$5 x^{2} + 9 x + 4 - 2 = 0$

Subtract $2$ from $4$ .

$5 x^{2} + 9 x + 2 = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = 5$ , $b = 9$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{- 9 \pm \sqrt{9^{2} - 4 \cdot (5 \cdot 2)}}{2 \cdot 5}$

Step 5

Simplify.

Simplify the numerator.

Raise $9$ to the power of $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 5 \cdot 2}}{2 \cdot 5}$

Multiply $- 4 \cdot 5 \cdot 2$ .

Multiply $- 4$ by $5$ .

$x = \frac{- 9 \pm \sqrt{81 - 20 \cdot 2}}{2 \cdot 5}$

Multiply $- 20$ by $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 40}}{2 \cdot 5}$

Subtract $40$ from $81$ .

$x = \frac{- 9 \pm \sqrt{41}}{2 \cdot 5}$

Multiply $2$ by $5$ .

$x = \frac{- 9 \pm \sqrt{41}}{10}$

Step 6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $9$ to the power of $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 5 \cdot 2}}{2 \cdot 5}$

Multiply $- 4 \cdot 5 \cdot 2$ .

Multiply $- 4$ by $5$ .

$x = \frac{- 9 \pm \sqrt{81 - 20 \cdot 2}}{2 \cdot 5}$

Multiply $- 20$ by $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 40}}{2 \cdot 5}$

Subtract $40$ from $81$ .

$x = \frac{- 9 \pm \sqrt{41}}{2 \cdot 5}$

Multiply $2$ by $5$ .

$x = \frac{- 9 \pm \sqrt{41}}{10}$

Change the $\pm$ to $+$ .

$x = \frac{- 9 + \sqrt{41}}{10}$

Rewrite $- 9$ as $- 1 (9)$ .

$x = \frac{- 1 \cdot 9 + \sqrt{41}}{10}$

Factor $- 1$ out of $\sqrt{41}$ .

$x = \frac{- 1 \cdot 9 - 1 (- \sqrt{41})}{10}$

Factor $- 1$ out of $- 1 (9) - 1 (- \sqrt{41})$ .

$x = \frac{- 1 (9 - \sqrt{41})}{10}$

Move the negative in front of the fraction.

$x = - \frac{9 - \sqrt{41}}{10}$

Step 7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $9$ to the power of $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 5 \cdot 2}}{2 \cdot 5}$

Multiply $- 4 \cdot 5 \cdot 2$ .

Multiply $- 4$ by $5$ .

$x = \frac{- 9 \pm \sqrt{81 - 20 \cdot 2}}{2 \cdot 5}$

Multiply $- 20$ by $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 40}}{2 \cdot 5}$

Subtract $40$ from $81$ .

$x = \frac{- 9 \pm \sqrt{41}}{2 \cdot 5}$

Multiply $2$ by $5$ .

$x = \frac{- 9 \pm \sqrt{41}}{10}$

Change the $\pm$ to $-$ .

$x = \frac{- 9 - \sqrt{41}}{10}$

Rewrite $- 9$ as $- 1 (9)$ .

$x = \frac{- 1 \cdot 9 - \sqrt{41}}{10}$

Factor $- 1$ out of $- \sqrt{41}$ .

$x = \frac{- 1 \cdot 9 - (\sqrt{41})}{10}$

Factor $- 1$ out of $- 1 (9) - (\sqrt{41})$ .

$x = \frac{- 1 (9 + \sqrt{41})}{10}$

Move the negative in front of the fraction.

$x = - \frac{9 + \sqrt{41}}{10}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{9 - \sqrt{41}}{10}, - \frac{9 + \sqrt{41}}{10}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{9 - \sqrt{41}}{10}, - \frac{9 + \sqrt{41}}{10}$

Decimal Form:

$x = - 0.25968757 \dots, - 1.54031242 \dots$

Solve by Factoring (5x+4)(x+1)=2

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