Solve (t-u)^2+5(3t-u+4u^2)(1+u)

$$( t - u ) ^ { 2 } + 5 ( 3 t - u + 4 u ^ { 2 } ) ( 1 + u )$$

Evaluate

$$5\left(u+1\right)\left(3t+4u^{2}-u\right)+\left(t-u\right)^{2}$$

Solution Steps

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

$$t^{2}-2tu+u^{2}+5\left(3t-u+4u^{2}\right)\left(1+u\right)$$

Use the distributive property to multiply 5 by 3t-u+4u^{2}.

$$t^{2}-2tu+u^{2}+\left(15t-5u+20u^{2}\right)\left(1+u\right)$$

Use the distributive property to multiply 15t-5u+20u^{2} by 1+u and combine like terms.

$$t^{2}-2tu+u^{2}+15t+15tu-5u+15u^{2}+20u^{3}$$

Combine -2tu and 15tu to get 13tu.

$$t^{2}+13tu+u^{2}+15t-5u+15u^{2}+20u^{3}$$

Combine u^{2} and 15u^{2} to get 16u^{2}.

$$t^{2}+13tu+16u^{2}+15t-5u+20u^{3}$$

Expand

$$t^{2}+13tu+15t+20u^{3}+16u^{2}-5u$$

Solution Steps

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

$$t^{2}-2tu+u^{2}+5\left(3t-u+4u^{2}\right)\left(1+u\right)$$

Use the distributive property to multiply 5 by 3t-u+4u^{2}.

$$t^{2}-2tu+u^{2}+\left(15t-5u+20u^{2}\right)\left(1+u\right)$$

Use the distributive property to multiply 15t-5u+20u^{2} by 1+u and combine like terms.

$$t^{2}-2tu+u^{2}+15t+15tu-5u+15u^{2}+20u^{3}$$

Combine -2tu and 15tu to get 13tu.

$$t^{2}+13tu+u^{2}+15t-5u+15u^{2}+20u^{3}$$

Combine u^{2} and 15u^{2} to get 16u^{2}.

$$t^{2}+13tu+16u^{2}+15t-5u+20u^{3}$$