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

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

Solve for M

$$M=\left(x+1\right)\left(5x+9\right)$$

Solution Steps

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

$$M=9x^{2}+30x+25-4\left(x+2\right)^{2}$$

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

$$M=9x^{2}+30x+25-4\left(x^{2}+4x+4\right)$$

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

$$M=9x^{2}+30x+25-4x^{2}-16x-16$$

Combine 9x^{2} and -4x^{2} to get 5x^{2}.

$$M=5x^{2}+30x+25-16x-16$$

Combine 30x and -16x to get 14x.

$$M=5x^{2}+14x+25-16$$

Subtract 16 from 25 to get 9.

$$M=5x^{2}+14x+9$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{5M+4}-7}{5}$$

$$x=\frac{\sqrt{5M+4}-7}{5}$$

Solve for x

$$x=\frac{-\sqrt{5M+4}-7}{5}$$

$$x=\frac{\sqrt{5M+4}-7}{5}\text{, }M\geq -\frac{4}{5}$$