Solve for x (x^2+4x+10)^(3/2)+1=0

${(x^{2} + 4 x + 10)}^{\frac{3}{2}} + 1 = 0$

Subtract $1$ from both sides of the equation.

${(x^{2} + 4 x + 10)}^{\frac{3}{2}} = - 1$

Raise each side of the equation to the power of $\frac{2}{3}$ to eliminate the fractional exponent on the left side.

${({(x^{2} + 4 x + 10)}^{\frac{3}{2}})}^{\frac{2}{3}} = {(- 1)}^{\frac{2}{3}}$

Simplify the exponent.

Simplify the left side.

Simplify ${({(x^{2} + 4 x + 10)}^{\frac{3}{2}})}^{\frac{2}{3}}$ .

Multiply the exponents in ${({(x^{2} + 4 x + 10)}^{\frac{3}{2}})}^{\frac{2}{3}}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x^{2} + 4 x + 10)}^{\frac{3}{2} \cdot \frac{2}{3}} = {(- 1)}^{\frac{2}{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x^{2} + 4 x + 10)}^{\frac{3}{2} \cdot \frac{2}{3}} = {(- 1)}^{\frac{2}{3}}$

Rewrite the expression.

${(x^{2} + 4 x + 10)}^{\frac{1}{2} \cdot 2} = {(- 1)}^{\frac{2}{3}}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x^{2} + 4 x + 10)}^{\frac{1}{2} \cdot 2} = {(- 1)}^{\frac{2}{3}}$

Rewrite the expression.

${(x^{2} + 4 x + 10)}^{1} = {(- 1)}^{\frac{2}{3}}$

Simplify.

$x^{2} + 4 x + 10 = {(- 1)}^{\frac{2}{3}}$

Simplify the right side.

Simplify ${(- 1)}^{\frac{2}{3}}$ .

Simplify the expression.

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$x^{2} + 4 x + 10 = {({(- 1)}^{3})}^{\frac{2}{3}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{2} + 4 x + 10 = {(- 1)}^{3 (\frac{2}{3})}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x^{2} + 4 x + 10 = {(- 1)}^{3 (\frac{2}{3})}$

Rewrite the expression.

$x^{2} + 4 x + 10 = {(- 1)}^{2}$

Raise $- 1$ to the power of $2$ .

$x^{2} + 4 x + 10 = 1$

Solve for $x$ .

Move all terms to the left side of the equation and simplify.

Subtract $1$ from both sides of the equation.

$x^{2} + 4 x + 10 - 1 = 0$

Subtract $1$ from $10$ .

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

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = 4$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot 9)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 9}}{2 \cdot 1}$

Multiply $- 4$ by $9$ .

$x = \frac{- 4 \pm \sqrt{16 - 36}}{2 \cdot 1}$

Subtract $36$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 20}}{2 \cdot 1}$

Rewrite $- 20$ as $- 1 (20)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 20}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (20)}$ as $\sqrt{- 1} \cdot \sqrt{20}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{20}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 4 \pm i \cdot \sqrt{20}}{2 \cdot 1}$

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$x = \frac{- 4 \pm i \cdot \sqrt{4 (5)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 4 \pm i \cdot \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (2 \sqrt{5})}{2 \cdot 1}$

Move $2$ to the left of $i$ .

$x = \frac{- 4 \pm 2 i \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2 i \sqrt{5}}{2}$

Simplify $\frac{- 4 \pm 2 i \sqrt{5}}{2}$ .

$x = - 2 \pm i \sqrt{5}$

The final answer is the combination of both solutions.

$x = - 2 + i \sqrt{5}, - 2 - i \sqrt{5}$

Solve for x (x^2+4x+10)^(3/2)+1=0

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