Solve 4x+left(x-5right)left(x+4right)+15=left(3-xright)left(x+3right)+left(x+6right)left(x-3right)

$$4x+ \left( x-5 \right) \left( x+4 \right) +15 = \left( 3-x \right) \left( x+3 \right) + \left( x+6 \right) \left( x-3 \right)$$

Solve for x (complex solution)

$$x=-2i$$

$$x=2i$$

Steps by Finding Square Root

Use the distributive property to multiply x-5 by x+4 and combine like terms.

$$4x+x^{2}-x-20+15=\left(3-x\right)\left(x+3\right)+\left(x+6\right)\left(x-3\right)$$

Combine 4x and -x to get 3x.

$$3x+x^{2}-20+15=\left(3-x\right)\left(x+3\right)+\left(x+6\right)\left(x-3\right)$$

Add -20 and 15 to get -5.

$$3x+x^{2}-5=\left(3-x\right)\left(x+3\right)+\left(x+6\right)\left(x-3\right)$$

Consider \left(3-x\right)\left(x+3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 3.

$$3x+x^{2}-5=9-x^{2}+\left(x+6\right)\left(x-3\right)$$

Use the distributive property to multiply x+6 by x-3 and combine like terms.

$$3x+x^{2}-5=9-x^{2}+x^{2}+3x-18$$

Combine -x^{2} and x^{2} to get 0.

$$3x+x^{2}-5=9+3x-18$$

Subtract 18 from 9 to get -9.

$$3x+x^{2}-5=-9+3x$$

Subtract 3x from both sides.

$$3x+x^{2}-5-3x=-9$$

Combine 3x and -3x to get 0.

$$x^{2}-5=-9$$

Add 5 to both sides.

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

Add -9 and 5 to get -4.

$$x^{2}=-4$$

The equation is now solved.

$$x=2i x=-2i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-5 by x+4 and combine like terms.

$$4x+x^{2}-x-20+15=\left(3-x\right)\left(x+3\right)+\left(x+6\right)\left(x-3\right)$$

Combine 4x and -x to get 3x.

$$3x+x^{2}-20+15=\left(3-x\right)\left(x+3\right)+\left(x+6\right)\left(x-3\right)$$

Add -20 and 15 to get -5.

$$3x+x^{2}-5=\left(3-x\right)\left(x+3\right)+\left(x+6\right)\left(x-3\right)$$

Consider \left(3-x\right)\left(x+3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 3.

$$3x+x^{2}-5=9-x^{2}+\left(x+6\right)\left(x-3\right)$$

Use the distributive property to multiply x+6 by x-3 and combine like terms.

$$3x+x^{2}-5=9-x^{2}+x^{2}+3x-18$$

Combine -x^{2} and x^{2} to get 0.

$$3x+x^{2}-5=9+3x-18$$

Subtract 18 from 9 to get -9.

$$3x+x^{2}-5=-9+3x$$

Subtract -9 from both sides.

$$3x+x^{2}-5-\left(-9\right)=3x$$

The opposite of -9 is 9.

$$3x+x^{2}-5+9=3x$$

Subtract 3x from both sides.

$$3x+x^{2}-5+9-3x=0$$

Add -5 and 9 to get 4.

$$3x+x^{2}+4-3x=0$$

Combine 3x and -3x to get 0.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 4}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 4}}{2}$$

Multiply -4 times 4.

$$x=\frac{0±\sqrt{-16}}{2}$$

Take the square root of -16.

$$x=\frac{0±4i}{2}$$

Now solve the equation x=\frac{0±4i}{2} when ± is plus.

$$x=2i$$

Now solve the equation x=\frac{0±4i}{2} when ± is minus.

$$x=-2i$$

The equation is now solved.

$$x=2i x=-2i$$