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

$$(5x+1)(2x-3)(5x+1)(x-5)+5x+1$$

Evaluate

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

Solution Steps

Multiply 5x+1 and 5x+1 to get \left(5x+1\right)^{2}.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5x+1\right)^{2}.

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

Apply the distributive property by multiplying each term of 25x^{2}+10x+1 by each term of 2x-3.

$$\left(50x^{3}-75x^{2}+20x^{2}-30x+2x-3\right)\left(x-5\right)+5x+1$$

Combine -75x^{2} and 20x^{2} to get -55x^{2}.

$$\left(50x^{3}-55x^{2}-30x+2x-3\right)\left(x-5\right)+5x+1$$

Combine -30x and 2x to get -28x.

$$\left(50x^{3}-55x^{2}-28x-3\right)\left(x-5\right)+5x+1$$

Apply the distributive property by multiplying each term of 50x^{3}-55x^{2}-28x-3 by each term of x-5.

$$50x^{4}-250x^{3}-55x^{3}+275x^{2}-28x^{2}+140x-3x+15+5x+1$$

Combine -250x^{3} and -55x^{3} to get -305x^{3}.

$$50x^{4}-305x^{3}+275x^{2}-28x^{2}+140x-3x+15+5x+1$$

Combine 275x^{2} and -28x^{2} to get 247x^{2}.

$$50x^{4}-305x^{3}+247x^{2}+140x-3x+15+5x+1$$

Combine 140x and -3x to get 137x.

$$50x^{4}-305x^{3}+247x^{2}+137x+15+5x+1$$

Combine 137x and 5x to get 142x.

$$50x^{4}-305x^{3}+247x^{2}+142x+15+1$$

Add 15 and 1 to get 16.

$$50x^{4}-305x^{3}+247x^{2}+142x+16$$

Expand

$$50x^{4}-305x^{3}+247x^{2}+142x+16$$

Solution Steps

Multiply 5x+1 and 5x+1 to get \left(5x+1\right)^{2}.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5x+1\right)^{2}.

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

Apply the distributive property by multiplying each term of 25x^{2}+10x+1 by each term of 2x-3.

$$\left(50x^{3}-75x^{2}+20x^{2}-30x+2x-3\right)\left(x-5\right)+5x+1$$

Combine -75x^{2} and 20x^{2} to get -55x^{2}.

$$\left(50x^{3}-55x^{2}-30x+2x-3\right)\left(x-5\right)+5x+1$$

Combine -30x and 2x to get -28x.

$$\left(50x^{3}-55x^{2}-28x-3\right)\left(x-5\right)+5x+1$$

Apply the distributive property by multiplying each term of 50x^{3}-55x^{2}-28x-3 by each term of x-5.

$$50x^{4}-250x^{3}-55x^{3}+275x^{2}-28x^{2}+140x-3x+15+5x+1$$

Combine -250x^{3} and -55x^{3} to get -305x^{3}.

$$50x^{4}-305x^{3}+275x^{2}-28x^{2}+140x-3x+15+5x+1$$

Combine 275x^{2} and -28x^{2} to get 247x^{2}.

$$50x^{4}-305x^{3}+247x^{2}+140x-3x+15+5x+1$$

Combine 140x and -3x to get 137x.

$$50x^{4}-305x^{3}+247x^{2}+137x+15+5x+1$$

Combine 137x and 5x to get 142x.

$$50x^{4}-305x^{3}+247x^{2}+142x+15+1$$

Add 15 and 1 to get 16.

$$50x^{4}-305x^{3}+247x^{2}+142x+16$$