Solve (x^2+5x)^2+10(x^2+5x)+25=

$$( x ^ { 2 } + 5 x ) ^ { 2 } + 10 ( x ^ { 2 } + 5 x ) + 25 =$$

Evaluate

$$\left(x^{2}+5x+5\right)^{2}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{2}+5x\right)^{2}.

$$\left(x^{2}\right)^{2}+10x^{2}x+25x^{2}+10\left(x^{2}+5x\right)+25$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{4}+10x^{2}x+25x^{2}+10\left(x^{2}+5x\right)+25$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$x^{4}+10x^{3}+25x^{2}+10\left(x^{2}+5x\right)+25$$

Use the distributive property to multiply 10 by x^{2}+5x.

$$x^{4}+10x^{3}+25x^{2}+10x^{2}+50x+25$$

Combine 25x^{2} and 10x^{2} to get 35x^{2}.

$$x^{4}+10x^{3}+35x^{2}+50x+25$$

Expand

$$x^{4}+10x^{3}+35x^{2}+50x+25$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{2}+5x\right)^{2}.

$$\left(x^{2}\right)^{2}+10x^{2}x+25x^{2}+10\left(x^{2}+5x\right)+25$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{4}+10x^{2}x+25x^{2}+10\left(x^{2}+5x\right)+25$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$x^{4}+10x^{3}+25x^{2}+10\left(x^{2}+5x\right)+25$$

Use the distributive property to multiply 10 by x^{2}+5x.

$$x^{4}+10x^{3}+25x^{2}+10x^{2}+50x+25$$

Combine 25x^{2} and 10x^{2} to get 35x^{2}.

$$x^{4}+10x^{3}+35x^{2}+50x+25$$