Solve for b 3x^3+6x^2-24x=ax(x+b)(x+c)

$3 x^{3} + 6 x^{2} - 24 x = a x (x + b) (x + c)$

Rewrite the equation as $a x (x + b) (x + c) = 3 x^{3} + 6 x^{2} - 24 x$ .

$a x (x + b) (x + c) = 3 x^{3} + 6 x^{2} - 24 x$

Divide each term in $a x (x + b) (x + c) = 3 x^{3} + 6 x^{2} - 24 x$ by $a x (x + c)$ and simplify.

Divide each term in $a x (x + b) (x + c) = 3 x^{3} + 6 x^{2} - 24 x$ by $a x (x + c)$ .

$\frac{a x (x + b) (x + c)}{a x (x + c)} = \frac{3 x^{3}}{a x (x + c)} + \frac{6 x^{2}}{a x (x + c)} + \frac{- 24 x}{a x (x + c)}$

Simplify the left side.

Cancel the common factor of $a$ .

Cancel the common factor.

$\frac{a x (x + b) (x + c)}{a x (x + c)} = \frac{3 x^{3}}{a x (x + c)} + \frac{6 x^{2}}{a x (x + c)} + \frac{- 24 x}{a x (x + c)}$

Rewrite the expression.

$\frac{(x (x + b)) (x + c)}{x (x + c)} = \frac{3 x^{3}}{a x (x + c)} + \frac{6 x^{2}}{a x (x + c)} + \frac{- 24 x}{a x (x + c)}$

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{x (x + b) (x + c)}{x (x + c)} = \frac{3 x^{3}}{a x (x + c)} + \frac{6 x^{2}}{a x (x + c)} + \frac{- 24 x}{a x (x + c)}$

Rewrite the expression.

$\frac{(x + b) (x + c)}{x + c} = \frac{3 x^{3}}{a x (x + c)} + \frac{6 x^{2}}{a x (x + c)} + \frac{- 24 x}{a x (x + c)}$

Cancel the common factor of $x + c$ .

Cancel the common factor.

$\frac{(x + b) (x + c)}{x + c} = \frac{3 x^{3}}{a x (x + c)} + \frac{6 x^{2}}{a x (x + c)} + \frac{- 24 x}{a x (x + c)}$

Divide $x + b$ by $1$ .

$x + b = \frac{3 x^{3}}{a x (x + c)} + \frac{6 x^{2}}{a x (x + c)} + \frac{- 24 x}{a x (x + c)}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $x^{3}$ and $x$ .

Factor $x$ out of $3 x^{3}$ .

$x + b = \frac{x (3 x^{2})}{a x (x + c)} + \frac{6 x^{2}}{a x (x + c)} + \frac{- 24 x}{a x (x + c)}$

Cancel the common factors.

Factor $x$ out of $a x (x + c)$ .

$x + b = \frac{x (3 x^{2})}{x (a (x + c))} + \frac{6 x^{2}}{a x (x + c)} + \frac{- 24 x}{a x (x + c)}$

Cancel the common factor.

$x + b = \frac{x (3 x^{2})}{x (a (x + c))} + \frac{6 x^{2}}{a x (x + c)} + \frac{- 24 x}{a x (x + c)}$

Rewrite the expression.

$x + b = \frac{3 x^{2}}{a (x + c)} + \frac{6 x^{2}}{a x (x + c)} + \frac{- 24 x}{a x (x + c)}$

Cancel the common factor of $x^{2}$ and $x$ .

Factor $x$ out of $6 x^{2}$ .

$x + b = \frac{3 x^{2}}{a (x + c)} + \frac{x (6 x)}{a x (x + c)} + \frac{- 24 x}{a x (x + c)}$

Cancel the common factors.

Factor $x$ out of $a x (x + c)$ .

$x + b = \frac{3 x^{2}}{a (x + c)} + \frac{x (6 x)}{x (a (x + c))} + \frac{- 24 x}{a x (x + c)}$

Cancel the common factor.

$x + b = \frac{3 x^{2}}{a (x + c)} + \frac{x (6 x)}{x (a (x + c))} + \frac{- 24 x}{a x (x + c)}$

Rewrite the expression.

$x + b = \frac{3 x^{2}}{a (x + c)} + \frac{6 x}{a (x + c)} + \frac{- 24 x}{a x (x + c)}$

Cancel the common factor of $x$ .

Cancel the common factor.

$x + b = \frac{3 x^{2}}{a (x + c)} + \frac{6 x}{a (x + c)} + \frac{- 24 x}{a x (x + c)}$

Rewrite the expression.

$x + b = \frac{3 x^{2}}{a (x + c)} + \frac{6 x}{a (x + c)} + \frac{- 24}{a (x + c)}$

Move the negative in front of the fraction.

$x + b = \frac{3 x^{2}}{a (x + c)} + \frac{6 x}{a (x + c)} - \frac{24}{a (x + c)}$

Subtract $x$ from both sides of the equation.

$b = \frac{3 x^{2}}{a (x + c)} + \frac{6 x}{a (x + c)} - \frac{24}{a (x + c)} - x$

Solve for b 3x^3+6x^2-24x=ax(x+b)(x+c)

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