Begin by rearranging the terms to group together the quadratic:

\(\displaystyle x^2-9-6xy+9y^2\)

\(\displaystyle (x^2-6xy+9y^2)-9\)

Now, convert the quadratic into a square:

\(\displaystyle (x-3y)^2-9\)

Finally, distribute the \(\displaystyle -9\):

\(\displaystyle (x-3y+3)(x-3y-3)\)

Begin by extracting \(\displaystyle mn\) from the polynomial:

\(\displaystyle m^4n+m^2n-mn^4-mn^2\)

\(\displaystyle mn(m^3+m-n^3-n)\)

Now, rearrange to combine like terms:

\(\displaystyle mn(m^3-n^3+m-n)\)

Extract the like terms and factor the cubic:

\(\displaystyle mn[(m-n)(m^2+mn+n^2)+1(m-n)]\)

Simplify by combining like terms:

\(\displaystyle mn(m-n)(m^2+mn+n^2+1)\)

Begin by extracting \(\displaystyle 3\) from the polynomial:

\(\displaystyle 3x^3+9x^2+9x+3\)

\(\displaystyle 3(x^3+3x^2+3x+1)\)

Now, rearrange to combine like terms:

\(\displaystyle 3(x^3+1+3x^2+3x)\)

Extract the like terms and factor the cubic:

\(\displaystyle 3[(x+1)(x^2-x+1)+3x(x+1)]\)

Simplify by combining like terms:

\(\displaystyle 3[(x+1)(x^2-x+1+3x)]\)

\(\displaystyle 3(x+1)(x^2+2x+1)\)

\(\displaystyle 3(x+1)(x+1)(x+1)\)

\(\displaystyle 3(x+1)^3\)

By the Rational Zeroes Theorem, any rational solution must be a factor of the constant, 6, divided by the factor of the leading coefficient, 14. 

Four of the answer choices have this characteristic:

\(\displaystyle 3 = 3 \div 1\)

\(\displaystyle \frac{1}{2} = 1\div 2\)

\(\displaystyle \frac{6}{7} = 6 \div 7\)

\(\displaystyle \frac{3}{14} =3 \div14\)

\(\displaystyle \frac{10}{3}\) is in lowest terms, and 3 is not a factor of 14. It is therefore the correct answer.

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.

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

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