Solve (x+5)^2(x+1)^3+(x+5)^3(x+1)^2

$$( x + 5 ) ^ { 2 } ( x + 1 ) ^ { 3 } + ( x + 5 ) ^ { 3 } ( x + 1 ) ^ { 2 }$$

Evaluate

$$2\left(x+3\right)\left(\left(x+1\right)\left(x+5\right)\right)^{2}$$

Solution Steps

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

$$\left(x^{2}+10x+25\right)\left(x+1\right)^{3}+\left(x+5\right)^{3}\left(x+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+1\right)^{3}.

$$\left(x^{2}+10x+25\right)\left(x^{3}+3x^{2}+3x+1\right)+\left(x+5\right)^{3}\left(x+1\right)^{2}$$

Use the distributive property to multiply x^{2}+10x+25 by x^{3}+3x^{2}+3x+1 and combine like terms.

$$x^{5}+13x^{4}+58x^{3}+106x^{2}+85x+25+\left(x+5\right)^{3}\left(x+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+5\right)^{3}.

$$x^{5}+13x^{4}+58x^{3}+106x^{2}+85x+25+\left(x^{3}+15x^{2}+75x+125\right)\left(x+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$x^{5}+13x^{4}+58x^{3}+106x^{2}+85x+25+\left(x^{3}+15x^{2}+75x+125\right)\left(x^{2}+2x+1\right)$$

Use the distributive property to multiply x^{3}+15x^{2}+75x+125 by x^{2}+2x+1 and combine like terms.

$$x^{5}+13x^{4}+58x^{3}+106x^{2}+85x+25+x^{5}+17x^{4}+106x^{3}+290x^{2}+325x+125$$

Combine x^{5} and x^{5} to get 2x^{5}.

$$2x^{5}+13x^{4}+58x^{3}+106x^{2}+85x+25+17x^{4}+106x^{3}+290x^{2}+325x+125$$

Combine 13x^{4} and 17x^{4} to get 30x^{4}.

$$2x^{5}+30x^{4}+58x^{3}+106x^{2}+85x+25+106x^{3}+290x^{2}+325x+125$$

Combine 58x^{3} and 106x^{3} to get 164x^{3}.

$$2x^{5}+30x^{4}+164x^{3}+106x^{2}+85x+25+290x^{2}+325x+125$$

Combine 106x^{2} and 290x^{2} to get 396x^{2}.

$$2x^{5}+30x^{4}+164x^{3}+396x^{2}+85x+25+325x+125$$

Combine 85x and 325x to get 410x.

$$2x^{5}+30x^{4}+164x^{3}+396x^{2}+410x+25+125$$

Add 25 and 125 to get 150.

$$2x^{5}+30x^{4}+164x^{3}+396x^{2}+410x+150$$

Expand

$$2x^{5}+30x^{4}+164x^{3}+396x^{2}+410x+150$$

Solution Steps

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

$$\left(x^{2}+10x+25\right)\left(x+1\right)^{3}+\left(x+5\right)^{3}\left(x+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+1\right)^{3}.

$$\left(x^{2}+10x+25\right)\left(x^{3}+3x^{2}+3x+1\right)+\left(x+5\right)^{3}\left(x+1\right)^{2}$$

Use the distributive property to multiply x^{2}+10x+25 by x^{3}+3x^{2}+3x+1 and combine like terms.

$$x^{5}+13x^{4}+58x^{3}+106x^{2}+85x+25+\left(x+5\right)^{3}\left(x+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+5\right)^{3}.

$$x^{5}+13x^{4}+58x^{3}+106x^{2}+85x+25+\left(x^{3}+15x^{2}+75x+125\right)\left(x+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$x^{5}+13x^{4}+58x^{3}+106x^{2}+85x+25+\left(x^{3}+15x^{2}+75x+125\right)\left(x^{2}+2x+1\right)$$

Use the distributive property to multiply x^{3}+15x^{2}+75x+125 by x^{2}+2x+1 and combine like terms.

$$x^{5}+13x^{4}+58x^{3}+106x^{2}+85x+25+x^{5}+17x^{4}+106x^{3}+290x^{2}+325x+125$$

Combine x^{5} and x^{5} to get 2x^{5}.

$$2x^{5}+13x^{4}+58x^{3}+106x^{2}+85x+25+17x^{4}+106x^{3}+290x^{2}+325x+125$$

Combine 13x^{4} and 17x^{4} to get 30x^{4}.

$$2x^{5}+30x^{4}+58x^{3}+106x^{2}+85x+25+106x^{3}+290x^{2}+325x+125$$

Combine 58x^{3} and 106x^{3} to get 164x^{3}.

$$2x^{5}+30x^{4}+164x^{3}+106x^{2}+85x+25+290x^{2}+325x+125$$

Combine 106x^{2} and 290x^{2} to get 396x^{2}.

$$2x^{5}+30x^{4}+164x^{3}+396x^{2}+85x+25+325x+125$$

Combine 85x and 325x to get 410x.

$$2x^{5}+30x^{4}+164x^{3}+396x^{2}+410x+25+125$$

Add 25 and 125 to get 150.

$$2x^{5}+30x^{4}+164x^{3}+396x^{2}+410x+150$$