Solve f(x)=(5x^2+4)^3(4x+5)

$$f ( x ) = ( 5 x ^ { 2 } + 4 ) ^ { 3 } ( 4 x + 5 )$$

Evaluate

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

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(5x^{2}+4\right)^{3}.

$$\left(125\left(x^{2}\right)^{3}+300\left(x^{2}\right)^{2}+240x^{2}+64\right)\left(4x+5\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$\left(125x^{6}+300\left(x^{2}\right)^{2}+240x^{2}+64\right)\left(4x+5\right)$$

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

$$\left(125x^{6}+300x^{4}+240x^{2}+64\right)\left(4x+5\right)$$

Use the distributive property to multiply 125x^{6}+300x^{4}+240x^{2}+64 by 4x+5.

$$500x^{7}+625x^{6}+1200x^{5}+1500x^{4}+960x^{3}+1200x^{2}+256x+320$$

Expand

$$500x^{7}+625x^{6}+1200x^{5}+1500x^{4}+960x^{3}+1200x^{2}+256x+320$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(5x^{2}+4\right)^{3}.

$$\left(125\left(x^{2}\right)^{3}+300\left(x^{2}\right)^{2}+240x^{2}+64\right)\left(4x+5\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$\left(125x^{6}+300\left(x^{2}\right)^{2}+240x^{2}+64\right)\left(4x+5\right)$$

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

$$\left(125x^{6}+300x^{4}+240x^{2}+64\right)\left(4x+5\right)$$

Use the distributive property to multiply 125x^{6}+300x^{4}+240x^{2}+64 by 4x+5.

$$500x^{7}+625x^{6}+1200x^{5}+1500x^{4}+960x^{3}+1200x^{2}+256x+320$$