Solve (2t+4)^{2}+(16t-9t^2)^2+(3t^2-4)^2

$$( 2 t + 4 ) ^ { 2 } + ( 16 t - 9 t ^ { 2 } ) ^ { 2 } + ( 3 t ^ { 2 } - 4 ) ^ { 2 }$$

Evaluate

$$90t^{4}-288t^{3}+236t^{2}+16t+32$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2t+4\right)^{2}.

$$4t^{2}+16t+16+\left(16t-9t^{2}\right)^{2}+\left(3t^{2}-4\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(16t-9t^{2}\right)^{2}.

$$4t^{2}+16t+16+256t^{2}-288tt^{2}+81\left(t^{2}\right)^{2}+\left(3t^{2}-4\right)^{2}$$

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

$$4t^{2}+16t+16+256t^{2}-288t^{3}+81\left(t^{2}\right)^{2}+\left(3t^{2}-4\right)^{2}$$

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

$$4t^{2}+16t+16+256t^{2}-288t^{3}+81t^{4}+\left(3t^{2}-4\right)^{2}$$

Combine 4t^{2} and 256t^{2} to get 260t^{2}.

$$260t^{2}+16t+16-288t^{3}+81t^{4}+\left(3t^{2}-4\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3t^{2}-4\right)^{2}.

$$260t^{2}+16t+16-288t^{3}+81t^{4}+9\left(t^{2}\right)^{2}-24t^{2}+16$$

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

$$260t^{2}+16t+16-288t^{3}+81t^{4}+9t^{4}-24t^{2}+16$$

Combine 81t^{4} and 9t^{4} to get 90t^{4}.

$$260t^{2}+16t+16-288t^{3}+90t^{4}-24t^{2}+16$$

Combine 260t^{2} and -24t^{2} to get 236t^{2}.

$$236t^{2}+16t+16-288t^{3}+90t^{4}+16$$

Add 16 and 16 to get 32.

$$236t^{2}+16t+32-288t^{3}+90t^{4}$$

Expand

$$90t^{4}-288t^{3}+236t^{2}+16t+32$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2t+4\right)^{2}.

$$4t^{2}+16t+16+\left(16t-9t^{2}\right)^{2}+\left(3t^{2}-4\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(16t-9t^{2}\right)^{2}.

$$4t^{2}+16t+16+256t^{2}-288tt^{2}+81\left(t^{2}\right)^{2}+\left(3t^{2}-4\right)^{2}$$

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

$$4t^{2}+16t+16+256t^{2}-288t^{3}+81\left(t^{2}\right)^{2}+\left(3t^{2}-4\right)^{2}$$

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

$$4t^{2}+16t+16+256t^{2}-288t^{3}+81t^{4}+\left(3t^{2}-4\right)^{2}$$

Combine 4t^{2} and 256t^{2} to get 260t^{2}.

$$260t^{2}+16t+16-288t^{3}+81t^{4}+\left(3t^{2}-4\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3t^{2}-4\right)^{2}.

$$260t^{2}+16t+16-288t^{3}+81t^{4}+9\left(t^{2}\right)^{2}-24t^{2}+16$$

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

$$260t^{2}+16t+16-288t^{3}+81t^{4}+9t^{4}-24t^{2}+16$$

Combine 81t^{4} and 9t^{4} to get 90t^{4}.

$$260t^{2}+16t+16-288t^{3}+90t^{4}-24t^{2}+16$$

Combine 260t^{2} and -24t^{2} to get 236t^{2}.

$$236t^{2}+16t+16-288t^{3}+90t^{4}+16$$

Add 16 and 16 to get 32.

$$236t^{2}+16t+32-288t^{3}+90t^{4}$$