Solve f(x)=3(x+4)^2(4x-1)(x-5)^2

$$f ( x ) = 3 ( x + 4 ) ^ { 2 } ( 4 x - 1 ) ( x - 5 ) ^ { 2 }$$

Evaluate

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

Solution Steps

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

$$3\left(x^{2}+8x+16\right)\left(4x-1\right)\left(x-5\right)^{2}$$

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

$$3\left(x^{2}+8x+16\right)\left(4x-1\right)\left(x^{2}-10x+25\right)$$

Use the distributive property to multiply 3 by x^{2}+8x+16.

$$\left(3x^{2}+24x+48\right)\left(4x-1\right)\left(x^{2}-10x+25\right)$$

Use the distributive property to multiply 3x^{2}+24x+48 by 4x-1 and combine like terms.

$$\left(12x^{3}+93x^{2}+168x-48\right)\left(x^{2}-10x+25\right)$$

Use the distributive property to multiply 12x^{3}+93x^{2}+168x-48 by x^{2}-10x+25 and combine like terms.

$$12x^{5}-27x^{4}-462x^{3}+597x^{2}+4680x-1200$$

Expand

$$12x^{5}-27x^{4}-462x^{3}+597x^{2}+4680x-1200$$

Solution Steps

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

$$3\left(x^{2}+8x+16\right)\left(4x-1\right)\left(x-5\right)^{2}$$

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

$$3\left(x^{2}+8x+16\right)\left(4x-1\right)\left(x^{2}-10x+25\right)$$

Use the distributive property to multiply 3 by x^{2}+8x+16.

$$\left(3x^{2}+24x+48\right)\left(4x-1\right)\left(x^{2}-10x+25\right)$$

Use the distributive property to multiply 3x^{2}+24x+48 by 4x-1 and combine like terms.

$$\left(12x^{3}+93x^{2}+168x-48\right)\left(x^{2}-10x+25\right)$$

Use the distributive property to multiply 12x^{3}+93x^{2}+168x-48 by x^{2}-10x+25 and combine like terms.

$$12x^{5}-27x^{4}-462x^{3}+597x^{2}+4680x-1200$$