What is the product?
$(4 s+2)\left(5 s^2+10 s+3\right)$
To find the product of the two polynomials \((4s + 2)\) and \((5s^2 + 10s + 3)\), we recognize this as an algebraic multiplication task. We will use the distributive property, which dictates that every term in the first polynomial must be multiplied by every term in the second polynomial. Specifically, we have:
\[(4s + 2)(5s^2 + 10s + 3) = 4s(5s^2 + 10s + 3) + 2(5s^2 + 10s + 3)\]
This expansion ensures that all possible combinations of terms are accounted for.
Now, we distribute each part separately:
First, multiply \(4s\) by each term inside the second parenthesis:
\[4s \times 5s^2 = 20s^3\]
\[4s \times 10s = 40s^2\]
\[4s \times 3 = 12s\]
Next, multiply \(2\) by each term inside the second parenthesis:
\[2 \times 5s^2 = 10s^2\]
\[2 \times 10s = 20s\]
\[2 \times 3 = 6\]
Summing these results gives us:
\[(20s^3 + 40s^2 + 12s) + (10s^2 + 20s + 6)\]
To finalize the expression, we combine the like terms (terms with the same power of \(s\)):
Putting it all together, we get:
\[20s^3 + 50s^2 + 32s + 6\]
Comparing this result to the given options, we identify that it matches option d.
Expanding polynomials is a foundational skill in algebra, commonly applied in fields like physics and engineering. For instance, when calculating the area of a shape with dimensions expressed as algebraic functions (such as a rectangular field with sides \(L=4s+2\) and \(W=5s^2+10s+3\)), you would multiply these expressions to find the total area \(A = L \times W\). Mastering this process ensures you can simplify complex relationships in design and modeling tasks efficiently.