1. Factor the equation (x + 2)(x + 3) = 0. 2. Set each factor equal to zero: x + 2 = 0 and x + 3 = 0. 3. Solve for x: x = -2 and x = -3.

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Rationalize the denominator of $\frac{2}{9+\sqrt{2}}$ by multiplying numerator and denominator by $9-\sqrt{2}$.

$$\frac{5}{8+\sqrt{2}+\frac{2\left(9-\sqrt{2}\right)}{\left(9+\sqrt{2}\right)\left(9-\sqrt{2}\right)}}$$

Consider $\left(9+\sqrt{2}\right)\left(9-\sqrt{2}\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$\frac{5}{8+\sqrt{2}+\frac{2\left(9-\sqrt{2}\right)}{9^{2}-\left(\sqrt{2}\right)^{2}}}$$

Square $9$. Square $\sqrt{2}$.

$$\frac{5}{8+\sqrt{2}+\frac{2\left(9-\sqrt{2}\right)}{81-2}}$$

Subtract $2$ from $81$ to get $79$.

$$\frac{5}{8+\sqrt{2}+\frac{2\left(9-\sqrt{2}\right)}{79}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply $8+\sqrt{2}$ times $\frac{79}{79}$.

$$\frac{5}{\frac{79\left(8+\sqrt{2}\right)}{79}+\frac{2\left(9-\sqrt{2}\right)}{79}}$$

Since $\frac{79\left(8+\sqrt{2}\right)}{79}$ and $\frac{2\left(9-\sqrt{2}\right)}{79}$ have the same denominator, add them by adding their numerators.

$$\frac{5}{\frac{79\left(8+\sqrt{2}\right)+2\left(9-\sqrt{2}\right)}{79}}$$

Do the multiplications in $79\left(8+\sqrt{2}\right)+2\left(9-\sqrt{2}\right)$.

$$\frac{5}{\frac{632+79\sqrt{2}+18-2\sqrt{2}}{79}}$$

Do the calculations in $632+79\sqrt{2}+18-2\sqrt{2}$.

$$\frac{5}{\frac{650+77\sqrt{2}}{79}}$$

Divide $5$ by $\frac{650+77\sqrt{2}}{79}$ by multiplying $5$ by the reciprocal of $\frac{650+77\sqrt{2}}{79}$.

$$\frac{5\times 79}{650+77\sqrt{2}}$$

Rationalize the denominator of $\frac{5\times 79}{650+77\sqrt{2}}$ by multiplying numerator and denominator by $650-77\sqrt{2}$.

$$\frac{5\times 79\left(650-77\sqrt{2}\right)}{\left(650+77\sqrt{2}\right)\left(650-77\sqrt{2}\right)}$$

Consider $\left(650+77\sqrt{2}\right)\left(650-77\sqrt{2}\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$\frac{5\times 79\left(650-77\sqrt{2}\right)}{650^{2}-\left(77\sqrt{2}\right)^{2}}$$

Multiply $5$ and $79$ to get $395$.

$$\frac{395\left(650-77\sqrt{2}\right)}{650^{2}-\left(77\sqrt{2}\right)^{2}}$$

Calculate $650$ to the power of $2$ and get $422500$.

$$\frac{395\left(650-77\sqrt{2}\right)}{422500-\left(77\sqrt{2}\right)^{2}}$$

Expand $\left(77\sqrt{2}\right)^{2}$.

$$\frac{395\left(650-77\sqrt{2}\right)}{422500-77^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate $77$ to the power of $2$ and get $5929$.

$$\frac{395\left(650-77\sqrt{2}\right)}{422500-5929\left(\sqrt{2}\right)^{2}}$$

The square of $\sqrt{2}$ is $2$.

$$\frac{395\left(650-77\sqrt{2}\right)}{422500-5929\times 2}$$

Multiply $5929$ and $2$ to get $11858$.

$$\frac{395\left(650-77\sqrt{2}\right)}{422500-11858}$$

Subtract $11858$ from $422500$ to get $410642$.

$$\frac{395\left(650-77\sqrt{2}\right)}{410642}$$

Divide $395\left(650-77\sqrt{2}\right)$ by $410642$ to get $\frac{5}{5198}\left(650-77\sqrt{2}\right)$.

$$\frac{5}{5198}\left(650-77\sqrt{2}\right)$$

Use the distributive property to multiply $\frac{5}{5198}$ by $650-77\sqrt{2}$.

$$\frac{5}{5198}\times 650+\frac{5}{5198}\left(-77\right)\sqrt{2}$$

Express $\frac{5}{5198}\times 650$ as a single fraction.

$$\frac{5\times 650}{5198}+\frac{5}{5198}\left(-77\right)\sqrt{2}$$

Multiply $5$ and $650$ to get $3250$.

$$\frac{3250}{5198}+\frac{5}{5198}\left(-77\right)\sqrt{2}$$

Reduce the fraction $\frac{3250}{5198}$ to lowest terms by extracting and canceling out $2$.

$$\frac{1625}{2599}+\frac{5}{5198}\left(-77\right)\sqrt{2}$$

Express $\frac{5}{5198}\left(-77\right)$ as a single fraction.

$$\frac{1625}{2599}+\frac{5\left(-77\right)}{5198}\sqrt{2}$$

Multiply $5$ and $-77$ to get $-385$.

$$\frac{1625}{2599}+\frac{-385}{5198}\sqrt{2}$$

Fraction $\frac{-385}{5198}$ can be rewritten as $-\frac{385}{5198}$ by extracting the negative sign.

$$\frac{1625}{2599}-\frac{385}{5198}\sqrt{2}$$

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