Solve (3x-4/3)(5x+1)-{left(5/3x+4right)}^2

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

Evaluate

$$\frac{110x^{2}}{9}-17x-\frac{52}{3}$$

Solution Steps

Use the distributive property to multiply 3x-\frac{4}{3} by 5x+1 and combine like terms.

$$15x^{2}-\frac{11}{3}x-\frac{4}{3}-\left(\frac{5}{3}x+4\right)^{2}$$

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

$$15x^{2}-\frac{11}{3}x-\frac{4}{3}-\left(\frac{25}{9}x^{2}+\frac{40}{3}x+16\right)$$

To find the opposite of \frac{25}{9}x^{2}+\frac{40}{3}x+16, find the opposite of each term.

$$15x^{2}-\frac{11}{3}x-\frac{4}{3}-\frac{25}{9}x^{2}-\frac{40}{3}x-16$$

Combine 15x^{2} and -\frac{25}{9}x^{2} to get \frac{110}{9}x^{2}.

$$\frac{110}{9}x^{2}-\frac{11}{3}x-\frac{4}{3}-\frac{40}{3}x-16$$

Combine -\frac{11}{3}x and -\frac{40}{3}x to get -17x.

$$\frac{110}{9}x^{2}-17x-\frac{4}{3}-16$$

Subtract 16 from -\frac{4}{3} to get -\frac{52}{3}.

$$\frac{110}{9}x^{2}-17x-\frac{52}{3}$$

Expand

$$\frac{110x^{2}}{9}-17x-\frac{52}{3}$$

Solution Steps

Use the distributive property to multiply 3x-\frac{4}{3} by 5x+1 and combine like terms.

$$15x^{2}-\frac{11}{3}x-\frac{4}{3}-\left(\frac{5}{3}x+4\right)^{2}$$

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

$$15x^{2}-\frac{11}{3}x-\frac{4}{3}-\left(\frac{25}{9}x^{2}+\frac{40}{3}x+16\right)$$

To find the opposite of \frac{25}{9}x^{2}+\frac{40}{3}x+16, find the opposite of each term.

$$15x^{2}-\frac{11}{3}x-\frac{4}{3}-\frac{25}{9}x^{2}-\frac{40}{3}x-16$$

Combine 15x^{2} and -\frac{25}{9}x^{2} to get \frac{110}{9}x^{2}.

$$\frac{110}{9}x^{2}-\frac{11}{3}x-\frac{4}{3}-\frac{40}{3}x-16$$

Combine -\frac{11}{3}x and -\frac{40}{3}x to get -17x.

$$\frac{110}{9}x^{2}-17x-\frac{4}{3}-16$$

Subtract 16 from -\frac{4}{3} to get -\frac{52}{3}.

$$\frac{110}{9}x^{2}-17x-\frac{52}{3}$$