Solve 5(3x^{2}+x)^2-2(x^3-x^2)^2

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

Evaluate

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

Solution Steps

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

$$5\left(9\left(x^{2}\right)^{2}+6x^{2}x+x^{2}\right)-2\left(x^{3}-x^{2}\right)^{2}$$

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

$$5\left(9x^{4}+6x^{2}x+x^{2}\right)-2\left(x^{3}-x^{2}\right)^{2}$$

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

$$5\left(9x^{4}+6x^{3}+x^{2}\right)-2\left(x^{3}-x^{2}\right)^{2}$$

Use the distributive property to multiply 5 by 9x^{4}+6x^{3}+x^{2}.

$$45x^{4}+30x^{3}+5x^{2}-2\left(x^{3}-x^{2}\right)^{2}$$

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

$$45x^{4}+30x^{3}+5x^{2}-2\left(\left(x^{3}\right)^{2}-2x^{3}x^{2}+\left(x^{2}\right)^{2}\right)$$

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

$$45x^{4}+30x^{3}+5x^{2}-2\left(x^{6}-2x^{3}x^{2}+\left(x^{2}\right)^{2}\right)$$

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

$$45x^{4}+30x^{3}+5x^{2}-2\left(x^{6}-2x^{5}+\left(x^{2}\right)^{2}\right)$$

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

$$45x^{4}+30x^{3}+5x^{2}-2\left(x^{6}-2x^{5}+x^{4}\right)$$

Use the distributive property to multiply -2 by x^{6}-2x^{5}+x^{4}.

$$45x^{4}+30x^{3}+5x^{2}-2x^{6}+4x^{5}-2x^{4}$$

Combine 45x^{4} and -2x^{4} to get 43x^{4}.

$$43x^{4}+30x^{3}+5x^{2}-2x^{6}+4x^{5}$$

Expand

$$5x^{2}+30x^{3}+43x^{4}+4x^{5}-2x^{6}$$

Solution Steps

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

$$5\left(9\left(x^{2}\right)^{2}+6x^{2}x+x^{2}\right)-2\left(x^{3}-x^{2}\right)^{2}$$

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

$$5\left(9x^{4}+6x^{2}x+x^{2}\right)-2\left(x^{3}-x^{2}\right)^{2}$$

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

$$5\left(9x^{4}+6x^{3}+x^{2}\right)-2\left(x^{3}-x^{2}\right)^{2}$$

Use the distributive property to multiply 5 by 9x^{4}+6x^{3}+x^{2}.

$$45x^{4}+30x^{3}+5x^{2}-2\left(x^{3}-x^{2}\right)^{2}$$

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

$$45x^{4}+30x^{3}+5x^{2}-2\left(\left(x^{3}\right)^{2}-2x^{3}x^{2}+\left(x^{2}\right)^{2}\right)$$

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

$$45x^{4}+30x^{3}+5x^{2}-2\left(x^{6}-2x^{3}x^{2}+\left(x^{2}\right)^{2}\right)$$

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

$$45x^{4}+30x^{3}+5x^{2}-2\left(x^{6}-2x^{5}+\left(x^{2}\right)^{2}\right)$$

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

$$45x^{4}+30x^{3}+5x^{2}-2\left(x^{6}-2x^{5}+x^{4}\right)$$

Use the distributive property to multiply -2 by x^{6}-2x^{5}+x^{4}.

$$45x^{4}+30x^{3}+5x^{2}-2x^{6}+4x^{5}-2x^{4}$$

Combine 45x^{4} and -2x^{4} to get 43x^{4}.

$$43x^{4}+30x^{3}+5x^{2}-2x^{6}+4x^{5}$$