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4x(8 - x)

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To simplify the expression \( 4x(8 - x) \), you can distribute \( 4x \) to both terms in the parentheses:

\[ 4x(8) - 4x(x) \]

This results in:

\[ 32x - 4x^2 \]

So, the simplified expression is:

\[ -4x^2 + 32x \]

Multiply the monomial and the polynomial.
3x to the 3rd power(3x + 4)

To multiply the monomial \( 3x^3 \) by the polynomial \( (3x + 4) \), you will distribute \( 3x^3 \) to each term in the polynomial.

  1. Multiply \( 3x^3 \) by \( 3x \): \[ 3x^3 \cdot 3x = 9x^{4} \]

  2. Multiply \( 3x^3 \) by \( 4 \): \[ 3x^3 \cdot 4 = 12x^{3} \]

Now, combine these results:

\[ 9x^{4} + 12x^{3} \]

So, the result of multiplying the monomial by the polynomial is:

\[ 9x^4 + 12x^3 \]

-5y to the 4th power(6y to the 2nd power + 7y - 2)

To multiply the monomial \(-5y^4\) by the polynomial \( (6y^2 + 7y - 2) \), you will distribute \(-5y^4\) to each term in the polynomial.

  1. Multiply \(-5y^4\) by \(6y^2\): \[ -5y^4 \cdot 6y^2 = -30y^{6} \]

  2. Multiply \(-5y^4\) by \(7y\): \[ -5y^4 \cdot 7y = -35y^{5} \]

  3. Multiply \(-5y^4\) by \(-2\): \[ -5y^4 \cdot -2 = 10y^{4} \]

Now, combine these results:

\[ -30y^6 - 35y^5 + 10y^4 \]

So, the result of multiplying the monomial by the polynomial is:

\[ -30y^6 - 35y^5 + 10y^4 \]

(5x - 3)(3x + 2)

To multiply the two binomials \( (5x - 3)(3x + 2) \), you can use the distributive property (often called the FOIL method for binomials):

  1. First: Multiply the first terms in each binomial: \[ 5x \cdot 3x = 15x^2 \]

  2. Outer: Multiply the outer terms: \[ 5x \cdot 2 = 10x \]

  3. Inner: Multiply the inner terms: \[ -3 \cdot 3x = -9x \]

  4. Last: Multiply the last terms: \[ -3 \cdot 2 = -6 \]

Now, combine all these results: \[ 15x^2 + 10x - 9x - 6 \]

Combine like terms (\(10x - 9x\)): \[ 15x^2 + (10x - 9x) - 6 = 15x^2 + x - 6 \]

So, the result of multiplying the binomials is:

\[ \boxed{15x^2 + x - 6} \]

what is the perimeter of 5s to the 2nd power + 8s - 12 + 18s to the 2nd power - 3s + 10 + s - 3

To find the perimeter, we first need to simplify the given expression. The expression is:

\[ 5s^2 + 8s - 12 + 18s^2 - 3s + 10 + s - 3 \]

We will combine like terms:

  1. Combine the \(s^2\) terms: \[ 5s^2 + 18s^2 = 23s^2 \]

  2. Combine the \(s\) terms: \[ 8s - 3s + s = 6s \]

  3. Combine the constant terms: \[ -12 + 10 - 3 = -5 \]

Now, putting it all together, we get:

\[ 23s^2 + 6s - 5 \]

If you are referring to "perimeter" in the context of a shape defined by the polynomial, the polynomial may not directly represent a geometric shape with a perimeter. Typically, perimeter Calculation applies to polygons or figures.

However, if you are looking for the expression for the perimeter as a function of \(s\), the result is:

\[ \boxed{23s^2 + 6s - 5} \]

If you meant something else by "perimeter," please clarify!