Solve v*r(x)=(4x^2+7x)^3

$$v \cdot r ( x ) = ( 4 x ^ { 2 } + 7 x ) ^ { 3 }$$

Solve for r

$$\left\{\begin{matrix}r=\frac{x^{2}\left(4x+7\right)^{3}}{v}\text{, }&v\neq 0\\r\in \mathrm{R}\text{, }&\left(x=-\frac{7}{4}\text{ and }v=0\right)\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(4x^{2}+7x\right)^{3}.

$$vrx=64\left(x^{2}\right)^{3}+336\left(x^{2}\right)^{2}x+588x^{2}x^{2}+343x^{3}$$

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

$$vrx=64x^{6}+336\left(x^{2}\right)^{2}x+588x^{2}x^{2}+343x^{3}$$

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

$$vrx=64x^{6}+336x^{4}x+588x^{2}x^{2}+343x^{3}$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$vrx=64x^{6}+336x^{5}+588x^{2}x^{2}+343x^{3}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$vrx=64x^{6}+336x^{5}+588x^{4}+343x^{3}$$

The equation is in standard form.

$$vxr=64x^{6}+336x^{5}+588x^{4}+343x^{3}$$

Divide both sides by vx.

$$\frac{vxr}{vx}=\frac{x^{3}\left(4x+7\right)^{3}}{vx}$$

Dividing by vx undoes the multiplication by vx.

$$r=\frac{x^{3}\left(4x+7\right)^{3}}{vx}$$

Divide \left(7+4x\right)^{3}x^{3} by vx.

$$r=\frac{x^{2}\left(4x+7\right)^{3}}{v}$$

Solve for v

$$\left\{\begin{matrix}v=\frac{x^{2}\left(4x+7\right)^{3}}{r}\text{, }&r\neq 0\\v\in \mathrm{R}\text{, }&\left(x=-\frac{7}{4}\text{ and }r=0\right)\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(4x^{2}+7x\right)^{3}.

$$vrx=64\left(x^{2}\right)^{3}+336\left(x^{2}\right)^{2}x+588x^{2}x^{2}+343x^{3}$$

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

$$vrx=64x^{6}+336\left(x^{2}\right)^{2}x+588x^{2}x^{2}+343x^{3}$$

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

$$vrx=64x^{6}+336x^{4}x+588x^{2}x^{2}+343x^{3}$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$vrx=64x^{6}+336x^{5}+588x^{2}x^{2}+343x^{3}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$vrx=64x^{6}+336x^{5}+588x^{4}+343x^{3}$$

The equation is in standard form.

$$rxv=64x^{6}+336x^{5}+588x^{4}+343x^{3}$$

Divide both sides by rx.

$$\frac{rxv}{rx}=\frac{x^{3}\left(4x+7\right)^{3}}{rx}$$

Dividing by rx undoes the multiplication by rx.

$$v=\frac{x^{3}\left(4x+7\right)^{3}}{rx}$$

Divide \left(7+4x\right)^{3}x^{3} by rx.

$$v=\frac{x^{2}\left(4x+7\right)^{3}}{r}$$