Factor the equation and solve:

$\displaystyle x^4-9=0$

$\displaystyle (x^2-3)(x^2+3)=0$

$\displaystyle x^2 = 3$

$\displaystyle x = \pm \sqrt{3}$

 or

$\displaystyle x^2 = -3$

$\displaystyle x = \pm i\sqrt{3}$

Therefore there are four answers:

$\displaystyle x=\pm\sqrt{3}, \pm i\sqrt{3}$

Factor the equation and solve:

$\displaystyle x-4\sqrt{x}-45=0$

$\displaystyle (\sqrt{x}-9)(\sqrt{x}+5)=0$

$\displaystyle \sqrt{x}=9$

$\displaystyle x=81$

or

$\displaystyle \sqrt{x}=-5$

This has no solutions.

Therefore there is only one answer:

$\displaystyle x=81$

In order to evaluate $\displaystyle \dpi{100} (2x+2y)^{2}$ one needs to multiply the expression by itself using the laws of FOIL.  In the foil method, one multiplies in the following order: first terms, outer terms, inner terms, and last terms.

$\displaystyle \dpi{100} \dpi{100} (2x+2y)^{2}=(2x+2y)(2x+2y)$

Multiply terms by way of FOIL method.

$\displaystyle =(2x*2x)+(2x*2y)+(2y*2x)+(2y*2y)$

Now multiply and simplify.

$\displaystyle =4x^{2}+4xy+4xy+4y^{2}$

$\displaystyle \rightarrow 4x^{2}+8xy+4y^{2}$

In order to evaluate $\displaystyle \dpi{100} \dpi{100} (x-2)^{2}$ one needs to multiply the expression by itself using the laws of FOIL.  In the foil method, one multiplies in the following order: first terms, outer terms, inner terms, and last terms.  Be sure to pay attention to signs.

$\displaystyle (x-2)^{2}=(x-2)(x-2)$

Multiply terms by way of FOIL method.

$\displaystyle =(x*x)+(x*-2)+(-2*x)+(-2*-2)$

Now multiply and simplify, paying attention to signs.

$\displaystyle =x^{2}+(-2x)+(-2x)+4$

$\displaystyle \rightarrow x^{2}-4x+4$

In order to evaluate $\displaystyle (x+2y)*(2x-y)$ one needs to multiply the expression by itself using the laws of FOIL.  In the foil method, one multiplies in the following order: first terms, outer terms, inner terms, and last terms.  Be sure to pay attention to signs.

Multiply terms by way of FOIL method.

$\displaystyle =(x*2x)+(x*-y)+(2y*2x)+(2y*-y)$

Now multiply and simplify, paying attention to signs.

$\displaystyle =2x^{2}+(-xy)+4xy+(-2y^{2})$

$\displaystyle \rightarrow 2x^{2}+3xy-2y^{2}$

In order to evaluate $\displaystyle \dpi{100} \dpi{100} (x-y)^{2}$ one needs to multiply the expression by itself using the laws of FOIL.  In the foil method, one multiplies in the following order: first terms, outer terms, inner terms, and last terms.  Be sure to pay attention to signs.

$\displaystyle (x-y)^{2}=(x-y)(x-y)$

Multiply terms by way of FOIL method.

$\displaystyle =(x*x)+(x*-y)+(-y*x)+(-y*-y)$

Now multiply and simplify, paying attention to signs.

$\displaystyle =x^{2}+(-xy)+(-xy)+(y^{2})$

$\displaystyle \rightarrow x^{2}-2xy+y^{2}$

Remember FOIL stands for First Outer Inner Last. That means we can take $\displaystyle (x+2)(x-2)$ and turn it into $\displaystyle x^2+2x-2x-4$.

Simplify to get $\displaystyle x^2-4$.

$\displaystyle (x+5)^2$ is the same thing as $\displaystyle (x+5)(x+5)$.

Remember that FOIL stands for First Outer Inner Last.

For this problem, that would be $\displaystyle x^2+5x+5x+25$. 

Simplify that to $\displaystyle x^2+10x+25$.

Remember that FOIL stands for First Outer Inner Last.

For this problem that would give us:

$\displaystyle (x+2)(x-2)$

$\displaystyle =x^2-2x+2x-4$

Simplify:

$\displaystyle =x^2-4$